Ewan Davies

Upcoming events

18–19 May 2022

DIMACS Workshop on Entropy and Optimization | New Brunswick, NJ (Online)

Preprints

Algorithms for the Ferromagnetic Potts Model on Expanders
C. Carlson, E. Davies, N. Fraiman, A. Kolla, A. Potukuchi, C. Yap
@article{CDF+22, title = {Algorithms for the Ferromagnetic {P}otts Model on Expanders}, author = {Carlson, Charlie and Davies, Ewan and Fraiman, Nicolas and Kolla, Alexandra and Potukuchi, Aditya and Yap, Corrine}, year = {2022}, month = apr, eprint = {2204.01923}, eprinttype = {arxiv}, primaryclass = {cs, math}, archiveprefix = {arXiv} }
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We give algorithms for approximating the partition function of the ferromagnetic Potts model on $d$ -regular expanding graphs. We require much weaker expansion than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models, using extremal graph theory and applications of Karger’s algorithm to counting cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We believe that these methods can shed more light on other important problems such as sub-exponential algorithms for approximate counting problems.
Packing list-colourings
S. Cambie, W. Cames van Batenburg, E. Davies, R.J. Kang
@article{CCDK21, title = {Packing list-colourings}, author = {Cambie, Stijn and {Cames van Batenburg}, Wouter and Davies, Ewan and Kang, Ross J.}, year = {2021}, month = oct, eprint = {2110.05230}, eprinttype = {arxiv}, primaryclass = {cs, math}, archiveprefix = {arXiv} }
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List colouring is an influential and classic topic in graph theory. We initiate the study of a natural strengthening of this problem, where instead of one list-colouring, we seek many in parallel. Our explorations have uncovered a potentially rich seam of interesting problems spanning chromatic graph theory. Given a $k$ -list-assignment $L$ of a graph $G$ , which is the assignment of a list $L(v)$ of $k$ colours to each vertex $v∈V(G)$ , we study the existence of $k$ pairwise-disjoint proper colourings of $G$ using colours from these lists. We may refer to this as a list-packing. Using a mix of combinatorial and probabilistic methods, we set out some basic upper bounds on the smallest $k$ for which such a list-packing is always guaranteed, in terms of the number of vertices, the degeneracy, the maximum degree, or the (list) chromatic number of $G$ . (The reader might already find it interesting that such a minimal $k$ is well defined.) We also pursue a more focused study of the case when $G$ is a bipartite graph. Our results do not yet rule out the tantalising prospect that the minimal $k$ above is not too much larger than the list chromatic number. Our study has taken inspiration from study of the strong chromatic number, and we also explore generalisations of the problem above in the same spirit.
An algorithmic framework for colouring locally sparse graphs
E. Davies, R.J. Kang, F. Pirot, J.-S. Sereni
@article{DKPS20a, title = {An algorithmic framework for colouring locally sparse graphs}, author = {Davies, Ewan and Kang, Ross J. and Pirot, Fran\c{c}ois and Sereni, Jean-S\'ebastien}, year = {2020}, eprinttype = {arxiv}, eprint = {2004.07151} }
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We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets.
With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$ , a randomised polynomial-time algorithm for colouring graphs of maximum degree $∆$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$ , where $1/2\le t\le ∆^\frac{2\varepsilon}{1+2\varepsilon}/(\log∆)^2$ , with $⌊(1+\varepsilon)∆/\log(∆/\sqrtt)⌋$ colours.
This generalises and improves upon several notable results including those of Kim (1995) and Alon, Krivelevich and Sudakov (1999), and more recent ones of Molloy (2019) and Achlioptas, Iliopoulos and Sinclair (2019). This bound on the chromatic number is tight up to an asymptotic factor $2$ and it coincides with a famous algorithmic barrier to colouring random graphs.
Graph structure via local occupancy
E. Davies, R.J. Kang, F. Pirot, J.-S. Sereni
@article{DKPS20, title = {Graph structure via local occupancy}, author = {Davies, Ewan and Kang, Ross J. and Pirot, Fran\c{c}ois and Sereni, Jean-S\'ebastien}, year = {2020}, eprinttype = {arxiv}, eprint = {2003.14361} }
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The first author together with Jenssen, Perkins and Roberts (2017) recently showed how local properties of the hard-core model on triangle-free graphs guarantee the existence of large independent sets, of size matching the best-known asymptotics due to Shearer (1983). The present work strengthens this in two ways: first, by guaranteeing stronger graph structure in terms of colourings through applications of the Lovász local lemma; and second, by extending beyond triangle-free graphs in terms of local sparsity, treating for example graphs of bounded local edge density, of bounded local Hall ratio, and of bounded clique number. This generalises and improves upon much other earlier work, including that of Shearer (1995), Alon (1996) and Alon, Krivelevich and Sudakov (1999), and more recent results of Molloy (2019), Bernshteyn (2019) and Achlioptas, Iliopoulos and Sinclair (2019). Our results derive from a common framework built around the hard-core model. It pivots on a property we call local occupancy, giving a clean separation between the methods for deriving graph structure with probabilistic information and verifying the requisite probabilistic information itself.
Efficient algorithms for the Potts model on small-set expanders
C. Carlson, E. Davies, A. Kolla
@article{CDK20, title = {Efficient algorithms for the Potts model on small-set expanders}, author = {Carlson, Charlie and Davies, Ewan and Kolla, Alexandra}, year = {2020}, eprinttype = {arxiv}, eprint = {2003.01154} }
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We develop an approximation algorithm for the partition function of the ferromagnetic Potts model on graphs with a small-set expansion condition, and as a step in the argument we give a graph partitioning algorithm with expansion and minimum degree conditions on the subgraphs induced by each part. These results extend previous work of Jenssen, Keevash, and Perkins (2019) on the Potts model and related problems in expander graphs, and of Oveis Gharan and Trevisan (2014) on partitioning into expanders.
Regularity inheritance in hypergraphs
P. Allen, E. Davies, J. Skokan
@article{ADS19, title = {Regularity inheritance in hypergraphs}, author = {Allen, Peter and Davies, Ewan and Skokan, Jozef}, year = {2019}, eprinttype = {arxiv}, eprint = {1901.05955} }
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We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.

Publications

Computational Thresholds for the Fixed-Magnetization Ising Model
C. Carlson, E. Davies, A. Kolla, W. Perkins
STOC (2022), to appear
@article{CDKP21, title = {Computational Thresholds for the Fixed-Magnetization {I}sing Model}, author = {Carlson, Charlie and Davies, Ewan and Kolla, Alexandra and Perkins, Will}, journal = {STOC}, pages = {to appear}, year = {2022}, eprint = {2111.03033}, eprinttype = {arxiv} }
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The ferromagnetic Ising model is a model of a magnetic material and a central topic in statistical physics. It also plays a starring role in the algorithmic study of approximate counting: approximating the partition function of the ferromagnetic Ising model with uniform external field is tractable at all temperatures and on all graphs, due to the randomized algorithm of Jerrum and Sinclair.
Here we show that hidden inside the model are hard computational problems. For the class of bounded-degree graphs we find computational thresholds for the approximate counting and sampling problems for the ferromagnetic Ising model at fixed magnetization (that is, fixing the number of $+1$ and $-1$ spins).
In particular, letting $\beta_c(∆)$ denote the critical inverse temperature of the zero-field Ising model on the infinite $∆$ -regular tree, and $\eta_{∆,β,1}^+$ denote the mean magnetization of the zero-field $+$ measure on the infinite $∆$ -regular tree at inverse temperature $β$ , we prove, for the class of graphs of maximum degree $∆$ :
1. For $β< \beta_c(∆)$ there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for all magnetizations $η$ .
2. For $β> \beta_c(∆)$ , there is an FPRAS and efficient sampling scheme for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| >\eta_{∆,β,1}^+$ .
3. For $β> \beta_c(∆)$ , there is no FPRAS for the fixed-magnetization Ising model for magnetizations $η$ such that $|η| <\eta_{∆,β,1}^+$ unless NP=RP.
The $χ$ -Ramsey Problem for Triangle-Free Graphs
E. Davies, F. Illingworth
SIAM Journal on Discrete Mathematics (2022), 1124–1134
@article{DI22, title = {The $\chi$-Ramsey Problem for Triangle-Free Graphs}, author = {Davies, Ewan and Illingworth, Freddie}, journal = {SIAM Journal on Discrete Mathematics}, pages = {1124--1134}, year = {2022}, month = jun, publisher = {{Society for Industrial and Applied Mathematics}}, issn = {0895-4801}, doi = {10.1137/21M1437573}, eprint = {2107.12288}, eprinttype = {arxiv} }
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In 1967, Erdős asked for the greatest chromatic number, $f(n)$ , amongst all $n$ -vertex, triangle-free graphs. An observation of Erdős and Hajnal together with Shearer’s classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$ . We improve this bound by a factor $\sqrt{2}$ , as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot.
Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs
E. Davies, W. Perkins
48th International Colloquium on Automata, Languages, and Programming (ICALP 2021) 198, 62:1–62:18
@inproceedings{DP21, title = {{Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, author = {Davies, Ewan and Perkins, Will}, year = {2021}, pages = {62:1--62:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, isbn = {978-3-95977-195-5}, issn = {1868-8969}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f\"ur Informatik}, address = {Dagstuhl, Germany}, url = {https://drops.dagstuhl.de/opus/volltexte/2021/14131}, urn = {urn:nbn:de:0030-drops-141310}, doi = {10.4230/LIPIcs.ICALP.2021.62}, annote = {Keywords: approximate counting, independent sets, Markov chains}, eprinttype = {arxiv}, eprint = {2102.04984} }
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We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density $\alpha_c(∆)$ and provide (i) for $α< \alpha_c(∆)$ randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most $αn$ in $n$ -vertex graphs of maximum degree $∆$ ; and (ii) a proof that unless NP=RP, no such algorithms exist for $α>\alpha_c(∆)$ . The critical density is the occupancy fraction of the hard core model on the clique $K_{∆+1}$ at the uniqueness threshold on the infinite $∆$ -regular tree, giving $\alpha_c(∆)∼\frace{1+e}\frac{1}{∆}$ as $∆\to∞$ .
An Approximate Blow-up Lemma for Sparse Hypergraphs
P. Allen, J. Böttcher, E.K. Hng, J. Skokan, E. Davies
Procedia Computer Science 195, 394–403
@inproceedings{ABDHS21, title = {An Approximate Blow-up Lemma for Sparse Hypergraphs}, journal = {Procedia Computer Science}, author = {Allen, Peter and B\"ottcher, Julia and Hng, Eng Keat and Skokan, Jozef and Davies, Ewan}, year = {2021}, series = {Proceedings of the XI Latin and American Algorithms, Graphs and Optimization Symposium.}, volume = {195}, pages = {394--403}, issn = {1877-0509}, doi = {10.1016/j.procs.2021.11.048} }
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We obtain an approximate sparse hypergraph version of the blow-up lemma, showing that partite hypergraphs with sufficient regularity of small subgraph counts behave as if they were complete partite for the purpose of embedding bounded degree hypergraphs.
A proof of the Upper Matching Conjecture for large graphs
E. Davies, M. Jenssen, W. Perkins
Journal of Combinatorial Theory, Series B 151 (2021), 393–416
@article{DJP21, title = {A proof of the Upper Matching Conjecture for large graphs}, author = {Davies, Ewan and Jenssen, Matthew and Perkins, Will}, shortjournal = {J. Combin. Theory Ser. B}, journal = {Journal of Combinatorial Theory, Series B}, year = {2021}, volume = {151}, pages = {393--416}, issn = {0095-8956}, doi = {10.1016/j.jctb.2021.07.005}, eprinttype = {arxiv}, eprint = {2004.06695} }
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We prove that the ‘Upper Matching Conjecture’ of Friedland, Krop, and Markström and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every $d$ and every large enough $n$ divisible by $2d$ , a union of $n/(2d)$ copies of the complete $d$ -regular bipartite graph maximizes the number of independent sets and matchings of size $k$ for each $k$ over all $d$ -regular graphs on $n$ vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth.
Statistical Physics Approaches to Unique Games
M. Coulson, E. Davies, A. Kolla, V. Patel, G. Regts
35th Computational Complexity Conference (CCC 2020) 169, 13:1–13:27
@inproceedings{CDKPR20, author = {Coulson, Matthew and Davies, Ewan and Kolla, Alexandra and Patel, Viresh and Regts, Guus}, title = {{Statistical Physics Approaches to Unique Games}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {13:1--13:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, isbn = {978-3-95977-156-6}, issn = {1868-8969}, year = {2020}, eprinttype = {arxiv}, eprint = {1911.01504}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum f\"{u}r Informatik}, address = {Dagstuhl, Germany}, url = {https://drops.dagstuhl.de/opus/volltexte/2020/12565}, urn = {urn:nbn:de:0030-drops-125650}, doi = {10.4230/LIPIcs.CCC.2020.13}, annote = {Keywords: Unique Games Conjecture, approximation algorithm, Potts model, cluster expansion, polynomial interpolation} }
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We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the “yes” case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the “yes” case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.
Occupancy fraction, fractional colouring, and triangle fraction
E. Davies, R. de Joannis de Verclos, R.J. Kang, F. Pirot
Journal of Graph Theory 97 (2021), 557–568
@article{DJKP21, title = {Occupancy fraction, fractional colouring, and triangle fraction}, author = {Davies, Ewan and de Joannis de Verclos, R\'{e}mi and Kang, Ross J. and Pirot, Fran\c{c}ois}, shortjournal = {J. Graph Theory}, journal = {Journal of Graph Theory}, year = {2021}, volume = {97}, number = {4}, pages = {557--568}, issn = {1097-0118}, doi = {10.1002/jgt.22671}, eprinttype = {arxiv}, eprint = {1812.11152} }
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Given $\varepsilon>0$ , there exists $f_0$ such that, if $f_0 \le f \le ∆^2+1$ , then for any graph $G$ on $n$ vertices of maximum degree $∆$ in which the neighbourhood of every vertex in $G$ spans at most $∆^2/f$ edges, we have (i) an independent set drawn uniformly at random from $G$ has expected size at least $(1/2-\varepsilon)(n/∆)\log f$ ; and (ii) the fractional chromatic number of $G$ is at most $(2+\varepsilon)∆/\log f$ .
The asymptotic factors cannot in general be improved by more than a factor of $2$ . One may view these as stronger versions of results of Ajtai, Komlós and Szemerédi (1981) and Shearer (1983) on the independence number. The proofs of both use a tight analysis of the hard-core model.
On Zero-Free Regions for the Anti-Ferromagnetic Potts Model on Bounded-Degree Graphs
F. Bencs, E. Davies, V. Patel, G. Regts
Annales De l’Institut Henri Poincaré D 8 (2021), 459–489
@article{BDPR21, title = {On Zero-Free Regions for the Anti-Ferromagnetic Potts Model on Bounded-Degree Graphs}, journal = {Annales de l'Institut Henri Poincar\'e D}, author = {Bencs, Ferenc and Davies, Ewan and Patel, Viresh and Regts, Guus}, year = {2021}, volume = {8}, number = {3}, pages = {459--489}, issn = {2308-5827}, doi = {10.4171/AIHPD/108}, eprinttype = {arxiv}, eprint = {1812.07532} }
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We give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $∆∈\mathbbN$ and any $k≥3.02 ∆+1$ , there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that $\mathbfZ(G;k,w)≠0$ for any $w∈U$ and any graph $G$ of maximum degree at most $∆$ . For small values of $∆$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$ -colourings of graphs of small maximum degree.
Colouring triangle-free graphs with local list sizes
E. Davies, R. de Joannis de Verclos, R.J. Kang, F. Pirot
Random Structures & Algorithms 57 (2020), 730–744
@article{DJKP18a, title = {Colouring triangle-free graphs with local list sizes}, author = {Davies, Ewan and de Joannis de Verclos, Rémi and Kang, Ross J. and Pirot, François}, shortjournal = {Random Structures Algorithms}, journal = {Random Structures \& Algorithms}, volume = {57}, number = {3}, pages = {730-744}, year = {2020}, doi = {10.1002/rsa.20945}, eprinttype = {arxiv}, eprint = {1812.01534} }
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We prove two distinct and natural refinements of a recent breakthrough result of Molloy (and a follow-up work of Bernshteyn) on the (list) chromatic number of triangle-free graphs. In both our results, we permit the amount of colour made available to vertices of lower degree to be accordingly lower. One result concerns list colouring and correspondence colouring, while the other concerns fractional colouring. Our proof of the second illustrates the use of the hard-core model to prove a Johansson-type result, which may be of independent interest.
Counting proper colourings in 4-regular graphs via the Potts model
E. Davies
Electronic Journal of Combinatorics 25 (2018), P4.7
@article{Dav18, title = {Counting proper colourings in 4-regular graphs via the Potts model}, author = {Davies, Ewan}, shortjournal = {Electron. J. Combin.}, journal = {Electronic Journal of Combinatorics}, volume = {25}, number = {4}, pages = {P4.7}, year = {2018}, doi = {10.37236/7743}, eprinttype = {arxiv}, eprint = {1801.07547} }
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We give tight upper and lower bounds on the internal energy per particle in the antiferromagnetic $q$ -state Potts model on $4$ -regular graphs, for $q\ge 5$ . This proves the first case of a conjecture of the author, Perkins, Jenssen, and Roberts on extensions of their methods, and implies tight bounds on the antiferromagnetic Potts partition function.
The zero-temperature limit gives upper and lower bounds on the number of proper $q$ -colourings of $4$ -regular graphs, which almost proves the case $d=4$ of a conjecture of Galvin and Tetali. For any $q \ge 5$ we prove that the number of proper $q$ -colourings of a $4$ -regular graph is maximised by a union of $K_{4,4}$ ’s.
Tight bounds on the coefficients of partition functions via stability
E. Davies, M. Jenssen, W. Perkins, B. Roberts
Journal of Combinatorial Theory, Series A 160 (2018), 1–30
@article{davies2017tight, title = {Tight bounds on the coefficients of partition functions via stability}, author = {Davies, Ewan and Jenssen, Matthew and Perkins, Will and Roberts, Barnaby}, shortjournal = {J. Combin. Theory Ser. A}, journal = {Journal of Combinatorial Theory, Series A}, volume = {160}, pages = {1--30}, year = {2018}, doi = {10.1016/j.jcta.2018.06.005}, eprinttype = {arxiv}, eprint = {1704.07784} }
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Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model and monomer-dimer model are the independence and matching polynomials respectively.
We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions.
As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of $q$ -colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least $5$ .
Extremes of the internal energy of the Potts model on cubic graphs
E. Davies, M. Jenssen, W. Perkins, B. Roberts
Random Structures & Algorithms 53 (2018), 59–75
@article{davies2016extremes, title = {Extremes of the internal energy of the {P}otts model on cubic graphs}, author = {Davies, Ewan and Jenssen, Matthew and Perkins, Will and Roberts, Barnaby}, shortjournal = {Random Structures Algorithms}, journal = {Random Structures \& Algorithms}, volume = {53}, number = {1}, pages = {59-75}, year = {2018}, doi = {10.1002/rsa.20767}, eprinttype = {arxiv}, eprint = {1610.08496} }
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We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$ . This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function.
Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$ -colorings of a $3$ -regular graph, for any $q \ge 2$ , is maximized by a union of $K_{3,3}$ ’s. This proves the $d=3$ case of a conjecture of Galvin and Tetali.
On the average size of independent sets in triangle-free graphs
E. Davies, M. Jenssen, W. Perkins, B. Roberts
Proceedings of the American Mathematical Society 146 (2018), 111–124
@article{davies2018average, title = {On the average size of independent sets in triangle-free graphs}, author = {Davies, Ewan and Jenssen, Matthew and Perkins, Will and Roberts, Barnaby}, shortjournal = {Proc. Amer. Math. Soc.}, journal = {Proceedings of the American Mathematical Society}, volume = {146}, pages = {111--124}, year = {2018}, doi = {10.1090/proc/13728}, eprinttype = {arxiv}, eprint = {1606.01043} }
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We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$ , showing that an independent set drawn uniformly at random from such a graph has expected size at least $(1+o_d(1)) \frac{\log d}dn$ . This gives an alternative proof of Shearer’s upper bound on the Ramsey number $R(3,k)$ . We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\exp \left[\left(\frac{1}{2}+o_d(1) \right) \frac{\log^2 d}dn \right]$ . The constant $1/2$ in the exponent is best possible. In both cases, tightness is exhibited by a random $d$ -regular graph.
Both results come from considering the hard-core model from statistical physics: a random independent set $I$ drawn from a graph with probability proportional to $λ^{|I|}$ , for a fugacity parameter $λ>0$ . We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree $d$ . The bound is asymptotically tight in $d$ for all $λ=O_d(1)$ .
We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory.
Multicolour Ramsey numbers of paths and even cycles
E. Davies, M. Jenssen, B. Roberts
European Journal of Combinatorics 63 (2017), 124–133
@article{davies2016multicolour, title = {Multicolour Ramsey numbers of paths and even cycles}, author = {Davies, Ewan and Jenssen, Matthew and Roberts, Barnaby}, shortjournal = {European J. Combin.}, journal = {European Journal of Combinatorics}, volume = {63}, pages = {124--133}, year = {2017}, doi = {10.1016/j.ejc.2017.03.002}, publisher = {Elsevier}, eprinttype = {arxiv}, eprint = {1606.00762} }
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We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)≤R_k(P_n)≤R_k(C_n)≤kn+o(n)$ . The upper bound was recently improved by Sárközy who showed that $R_k(C_n)≤\left(k-\frack{16k^3+1}\right)n+o(n)$ . Here we show $R_k(C_n) ≤(k-\frac14)n +o(n)$ , obtaining the first improvement to the coefficient of the linear term by an absolute constant.
Independent Sets, Matchings, and Occupancy Fractions
E. Davies, M. Jenssen, W. Perkins, B. Roberts
Journal of the London Mathematical Society 96 (2017), 47–66
@article{davies2015independent, title = {{Independent Sets, Matchings, and Occupancy Fractions}}, author = {Davies, Ewan and Jenssen, Matthew and Perkins, Will and Roberts, Barnaby}, shortjournal = {J. Lond. Math. Soc.}, journal = {Journal of the London Mathematical Society}, volume = {96}, pages = {47--66}, year = {2017}, doi = {10.1112/jlms.12056}, eprinttype = {arxiv}, eprint = {1508.04675} }
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We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of $d$ -regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of $K_{d,d}$ maximizes the number of independent sets and the independence polynomial of a $d$ -regular graph.
For matchings, this shows that the matching polynomial and the total number of matchings of a $d$ -regular graph are maximized by a union of copies of $K_{d,d}$ . Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström.
In probabilistic language, our main theorems state that for all $d$ -regular graphs and all $λ$ , the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity $λ$ are maximized by $K_{d,d}$ . Our method involves constrained optimization problems over distributions of random variables and applies to all $d$ -regular graphs directly, without a reduction to the bipartite case.
Tight bounds on the coefficients of partition functions via stability (extended abstract)
E. Davies, M. Jenssen, W. Perkins, B. Roberts
Electronic Notes in Discrete Mathematics 61 (2017), 317–321
@article{davies2017tightabs, title = {Tight bounds on the coefficients of partition functions via stability (extended abstract)}, author = {Davies, Ewan and Jenssen, Matthew and Perkins, Will and Roberts, Barnaby}, shortjournal = {Electron. Notes Discrete Math.}, journal = {Electronic Notes in Discrete Mathematics}, volume = {61}, pages = {317--321}, year = {2017}, doi = {10.1016/j.endm.2017.06.054}, publisher = {Elsevier} }
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We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.
As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of $q$ -colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least $5$ .
Counting in hypergraphs via regularity inheritance (extended abstract)
E. Davies
Electronic Notes in Discrete Mathematics 49 (2015), 413–417
@article{davies2015counting, title = {Counting in hypergraphs via regularity inheritance (extended abstract)}, author = {Davies, Ewan}, shortjournal = {Electron. Notes Discrete Math.}, journal = {Electronic Notes in Discrete Mathematics}, volume = {49}, pages = {413--417}, year = {2015}, doi = {10.1016/j.endm.2015.06.058}, publisher = {Elsevier} }
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We develop a theory of regularity inheritance in $3$ -uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma.

Other work

Extremal and probabilistic results for regular graphs
E. Davies
PhD thesis, The London School of Economics and Political Science (2017)
@phdthesis{davies2017extremal, title = {Extremal and probabilistic results for regular graphs}, author = {Davies, Ewan}, institution = {The London School of Economics and Political Science}, year = {2017}, doi = {10.21953/lse.bijk2dsj3hlb} }
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In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is $d$ -regularity, which means each vertex of a graph is contained in exactly $d$ edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness.
We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in $d$ -regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory.
We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs.
Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on $kn$ vertices are coloured with $k$ colours, one can find a monochromatic path on $n$ vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles.
Counting the number of ways a gas can fill a room
E. Davies
Maths at LSE Blog (2016)
@article{davies2016blog, title = {Counting the number of ways a gas can fill a room}, author = {Davies, Ewan}, journal = {Maths at LSE Blog}, year = {2016}, url = {http://blogs.lse.ac.uk/maths/2016/01/14/ewan-davies-counting-the-number-of-ways-a-gas-can-fill-a-room-3} }
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Graph theory is the study of connected systems of abstract ‘things’ which we call graphs. In this example the ‘things’ in the graph are particles of a gas or atoms in a molecule, and we describe a new method for understanding mathematical models in these graphs.

Past events

14–15 May 2022

AMS Spring Western Sectional Meeting | Denver, CO (Online)

Talk: Bounding the (list) chromatic number of triangle-free graphs (15 May, slides)
14–15 May 2022

7th Lake Michigan Workshop on Combinatorics and Graph Theory | Chicago, IL
27 Sep 2021

Combinatorics and Probability Seminar, UIC | Chicago, Illinois

Title: Graph coloring, independent transversals and packing
12–16 Jul 2021

ICALP 2021 | Glasgow, Scotland (Online)
17–21 May 2021

Extremal and algorithmic aspects of partition functions, Sparse Graphs Coalition | Online
8–12 Mar 2021

Entropy Compression and Related Methods, Sparse Graphs Coalition | Online

Talk: The cluster expansion, Lovász local lemma and algorithms (10 Mar)
19 Feb 2021

Discrete Math and Combinatorics Seminar, UofSC | Columbia, South Carolina (Online)

Title: Independent sets of a given size
8 Feb 2021

Combinatorics and Probability Seminar, UIC | Chicago, Illinois (Online)

Title: Independent sets of a given size
14–16 Dec 2020

Uniqueness methods in statistical mechanics | Online
21–25 Sep 2020

Computational Phase Transitions | Berkeley, California (Online)
8–11 Sep 2020

Geometry of Polynomials Reunion | Berkeley, California (Online)
19–28 Aug 2020

Probability, Geometry, and Computation in High Dimensions Boot Camp | Berkeley, California (Online)
28–31 Jul 2020

Computational Complexity Conference (CCC) 2020 | Saarbrücken, Germany (Online)

Talk: Statistical physics approaches to Unique Games (29 Jul)
4–5 Apr 2020

POSTPONED Lake Michigan Workshop on Combinatorics and Graph Theory | Chicago, Illinois
19 Mar 2020

POSTPONED Computer Science Theory Seminar, UIC | Chicago, Illinois

Title: Statistical physics approaches to Unique Games (with Alex Kolla)
12 Feb 2020

Discrete Mathematics Seminar, KdVI / CWI | Amsterdam, Netherlands

Title: Statistical physics approaches to Unique Games
22 Jan 2020

CS Theory Seminar, CU Boulder | Boulder, Colorado

Title: Statistical physics approaches to Unique Games
25 Sep 2019

CS Theory Seminar, CU Boulder | Boulder, Colorado

Title: Graph colouring and the hard-core model
15 Jan – 17 May 2019

Simons Institute: Geometry of Polynomials | Berkeley, California

Talk: Colouring locally sparse graphs via the hard-core model (17 Apr)
23–24 Aug 2018

Algorithmic and Combinatorial Aspects of Partition Functions | Amsterdam, Netherlands

Talk: Fractional colouring of triangle-free graphs via the hard-core model (24 Aug)
9–10 May 2018

Two one-day Colloquia in Combinatorics | London, UK
1 May 2018

Applied Stochastics Seminar, Radboud University | Nijmegen, Netherlands

Title: Using Gibbs measures in extremal combinatorics
26–30 Mar 2018

Workshop on Graph limits | Janov, Czechia

Talk: Tight bounds on the coefficients of partition functions via stability (30 Mar)
19 Feb – 4 Mar 2018

Graphs and Randomness, IMPA | Rio de Janeiro, Brazil
26 Jan 2018

Discrete Mathematics Seminar, KdVI / CWI | Amsterdam, Netherlands

Title: A new approach to Sidorenko's conjecture
18 Oct 2017

Discrete Mathematics Seminar, KdVI / CWI | Amsterdam, Netherlands

Title: Shamir's problem revisited
28 Aug – 1 Sep 2017

EUROCOMB 2017 | Vienna, Austria

Talk: Tight bounds on the coefficients of partition functions via stability (31 Aug, slides)
17–21 Jul 2017

Novi Sad Workshop on Foundations Of Computer Science | Novi Sad, Serbia

Talk: Regularity inheritance in 3-uniform hypergraphs (20 Jul)
11 May 2017

Two one-day Colloquia in Combinatorics (invited) | London, UK

Title: Tight bounds on the coefficients of partition functions via stability
21 Feb 2017

Combinatorics Seminar | Oxford, UK

Title: Extremal problems on colourings in cubic graphs via the Potts model
3 Feb 2017

Discrete Mathematics Seminar, KdVI / CWI | Amsterdam, Netherlands

Title: A probabilistic approach to bounding graph polynomials
27 Jan 2017

PhD Seminar on Combinatorics, Games and Optimisation, LSE | London, UK

Title: A probabilistic approach to bounding graph polynomials
3 Aug 2016

Student Combinatorics Day (invited) | Birmingham, UK

Title: On the average size of independent sets in triangle-free graphs (slides)
8 Apr 2016

Seminário de Teoria da Computação, Combinatória e Otimização | São Paulo, Brazil

Title: Independent sets, matchings, and occupancy fractions II
3 Nov 2015

Discrete Geometry and Combinatorics Seminar, UCL | London, UK

Title: Independent sets, matchings, and occupancy fractions
9 Oct 2015

Mathematics Lunchtime Seminar, LSE | London, UK

Title: Independent sets, matchings, and occupancy fractions I
31 Aug – 4 Sep 2015

EUROCOMB 2015 | Bergen, Norway

Talk: Counting in hypergraphs via regularity inheritance (31 Aug, slides)
15 Apr 2015

Postgraduate Combinatorial Conference, QMUL | London, UK

Title: Counting in hypergraphs via regularity inheritance
9 Apr 2015

Combinatorics Seminar, Freie Universität | Berlin, Germany

Title: Regularity inheritance in 3-uniform hypergraphs
3 Jul 2014

SUMMIT 190: Balogh, Csaba, Hajnal, and Pluhar are 190 (invited) | Szeged, Hungary

Title: Robustness of triangle factors
29 Nov 2013

Mathematics Lunchtime Seminar, LSE | London, UK

Title: Cycle packing

Current

None

Past

CSCI 3104: Algorithms, CU Boulder, Fall 2020
Representation Theory, University of Amsterdam, 2018
Algebraic Methods in Combinatorics, University of Amsterdam, 2018
Mathematics Support Centre, London School of Economics, 2015, 2016
MA203 Real Analysis, London School of Economics, 2015
MA316 Graph Theory, London School of Economics, 2014
MA212 Further Mathematical Methods, London School of Economics, 2014
MA100 Mathematical Methods, London School of Economics, 2013

View my full CV as a pdf here

Positions held

Postdoc at CU Boulder (2019-Present)
Research Fellow at the Simons Institute Geometry of Polynomials Program (2019)
Postdoc at the University of Amsterdam (2017-2018)

Education

Ph.D. Mathematics, London School of Economics and Political Science (2013-2017)
M.Math, University of Cambridge (2012-2013)
B.A. (Hons) Mathematics, University of Cambridge (2009-2012)

Awards

PhD Prize for Outstanding Academic Performance, London School of Economics
Mathematics Department New Teacher Prize, London School of Economics
Foundation Scholarship and R.A. Watchman Prize, Jesus College, Cambridge
Foundation Scholarship and Sir Harold Spencer Jones Prize, Jesus College, Cambridge
Foundation Scholarship and Ware Prize, Jesus College, Cambridge
Foundation Exhibition and Bronowski Prize, Jesus College, Cambridge

Some demonstrations of my work can be found here