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27 Sep 2021Combinatorics and Probability Seminar, UIC  Chicago, IllinoisTitle: Graph coloring, independent transversals and packing
Preprints
 The $χ$Ramsey Problem for TriangleFree GraphsE. Davies, F. Illingworth
@article{DI21, title = {The $\chi$Ramsey Problem for TriangleFree Graphs}, author = {Davies, Ewan and Illingworth, Freddie}, year = {2021}, eprinttype = {arxiv}, eprint = {2107.12288} }
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In 1967, Erdős asked for the greatest chromatic number, $f(n)$, amongst all $n$vertex, trianglefree graphs. An observation of Erdős and Hajnal together with Shearer’s classical upper bound for the offdiagonal 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.
 An algorithmic framework for colouring locally sparse graphsE. 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, JeanS\'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 polynomialtime 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 occupancyE. 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, JeanS\'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 hardcore model on trianglefree graphs guarantee the existence of large independent sets, of size matching the bestknown 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 trianglefree 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 hardcore 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 smallset expandersC. Carlson, E. Davies, A. Kolla
@article{CDK20, title = {Efficient algorithms for the Potts model on smallset expanders}, author = {Carlson, Charles 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 smallset 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 hypergraphsP. 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 vertexbyvertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.
Journal publications
 A proof of the Upper Matching Conjecture for large graphsE. Davies, M. Jenssen, W. PerkinsJournal 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}, month = nov, volume = {151}, pages = {393416}, issn = {00958956}, 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.
 Occupancy fraction, fractional colouring, and triangle fractionE. Davies, R. de Joannis de Verclos, R.J. Kang, F. PirotJournal 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 = {557568}, issn = {10970118}, 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 hardcore model.
 On zerofree regions for the antiferromagnetic Potts model on boundeddegree graphsF. Bencs, E. Davies, V. Patel, G. RegtsAnnales De l’Institut Henri Poincaré D (2021), to appear
@article{BDPR18, title = {On zerofree regions for the antiferromagnetic Potts model on boundeddegree graphs}, author = {Bencs, Ferenc and Davies, Ewan and Patel, Viresh and Regts, Guus}, shortjournal = {Ann. Inst. Henri Poincar\'e Comb. Phys. Interact.}, journal = {Annales de l'Institut Henri Poincar\'e D}, pages = {to appear}, year = {2021}, eprinttype = {arxiv}, eprint = {1812.07532} }
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We give zerofree regions for the partition function of the antiferromagnetic 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 trianglefree graphs with local list sizesE. Davies, R. de Joannis de Verclos, R.J. Kang, F. PirotRandom Structures & Algorithms 57 (2020), 730–744
@article{DJKP18a, title = {Colouring trianglefree 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 = {730744}, 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 followup work of Bernshteyn) on the (list) chromatic number of trianglefree 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 hardcore model to prove a Johanssontype result, which may be of independent interest.
 Counting proper colourings in 4regular graphs via the Potts modelE. DaviesElectronic Journal of Combinatorics 25 (2018), P4.7
@article{Dav18, title = {Counting proper colourings in 4regular 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 zerotemperature 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 stabilityE. Davies, M. Jenssen, W. Perkins, B. RobertsJournal 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 = {130}, 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 hardcore model and monomerdimer 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 graphsE. Davies, M. Jenssen, W. Perkins, B. RobertsRandom 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 = {5975}, 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 antiferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the antiferromagnetic Potts partition function.
Taking the zerotemperature 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 trianglefree graphsE. Davies, M. Jenssen, W. Perkins, B. RobertsProceedings of the American Mathematical Society 146 (2018), 111–124
@article{davies2018average, title = {On the average size of independent sets in trianglefree 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 = {111124}, 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 trianglefree 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 trianglefree 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 hardcore 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 hardcore model on trianglefree 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 trianglefree graph and give some consequences of these conjectures in Ramsey theory.
 Multicolour Ramsey numbers of paths and even cyclesE. Davies, M. Jenssen, B. RobertsEuropean 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 = {124133}, 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 $(k1)n+o(n)≤R_k(P_n)≤R_k(C_n)≤kn+o(n)$. 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 FractionsE. Davies, M. Jenssen, W. Perkins, B. RobertsJournal 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 = {4766}, 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 hardcore model and the edge occupancy fraction of the monomerdimer 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.
Conference publications
 Approximately Counting Independent Sets of a Given Size in BoundedDegree GraphsE. Davies, W. Perkins48th 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 BoundedDegree Graphs}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, author = {Davies, Ewan and Perkins, Will}, year = {2021}, pages = {62:162:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, isbn = {9783959771955}, issn = {18688969}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl  LeibnizZentrum f\"ur Informatik}, address = {Dagstuhl, Germany}, url = {https://drops.dagstuhl.de/opus/volltexte/2021/14131}, urn = {urn:nbn:de:0030drops141310}, 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 boundeddegree graphs. That is, we identify a critical density $\alpha_c(∆)$ and provide (i) for $α< \alpha_c(∆)$ randomized polynomialtime 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 blowup lemma for sparse hypergraphsP. Allen, J. Böttcher, E. Davies, E.K. Hng, J. SkokanLAGOS 2021, to appear
@inproceedings{ABDHS21, title = {An approximate blowup lemma for sparse hypergraphs}, journal = {LAGOS 2021}, author = {Allen, Peter and B\"ottcher, Julia and Davies, Ewan and Hng, Eng Keat and Skokan, Jozef}, year = {2021}, pages = {to appear} }
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 Statistical Physics Approaches to Unique GamesM. Coulson, E. Davies, A. Kolla, V. Patel, G. Regts35th 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 = {13:113:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, isbn = {9783959771566}, issn = {18688969}, year = {2020}, eprinttype = {arxiv}, eprint = {1911.01504}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss DagstuhlLeibnizZentrum f\"{u}r Informatik}, address = {Dagstuhl, Germany}, url = {https://drops.dagstuhl.de/opus/volltexte/2020/12565}, urn = {urn:nbn:de:0030drops125650}, 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 zerofree 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.
 Tight bounds on the coefficients of partition functions via stability (extended abstract)E. Davies, M. Jenssen, W. Perkins, B. RobertsElectronic 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 = {317321}, year = {2017}, doi = {10.1016/j.endm.2017.06.054}, publisher = {Elsevier} }
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We show how to use the recentlydeveloped 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. DaviesElectronic 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 = {413417}, 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 blowup lemma.
Other publications
 Extremal and probabilistic results for regular graphsE. DaviesPhD 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 hardcore 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 widelyapplied 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 vertexbyvertex, 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 wellknown 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 roomE. DaviesMaths 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/ewandaviescountingthenumberofwaysagascanfillaroom3} }
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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

12–16 Jul 2021ICALP 2021  Glasgow, Scotland (Online)

17–21 May 2021Extremal and algorithmic aspects of partition functions, Sparse Graphs Coalition  Online

8–12 Mar 2021Entropy Compression and Related Methods, Sparse Graphs Coalition  OnlineTalk: The cluster expansion, Lovász local lemma and algorithms (10 Mar)

19 Feb 2021Discrete Math and Combinatorics Seminar, UofSC  Columbia, South Carolina (Online)Title: Independent sets of a given size

8 Feb 2021Combinatorics and Probability Seminar, UIC  Chicago, Illinois (Online)Title: Independent sets of a given size

14–16 Dec 2020

21–25 Sep 2020Computational Phase Transitions  Berkeley, California (Online)

8–11 Sep 2020Geometry of Polynomials Reunion  Berkeley, California (Online)

19–28 Aug 2020Probability, Geometry, and Computation in High Dimensions Boot Camp  Berkeley, California (Online)

28–31 Jul 2020Computational Complexity Conference (CCC) 2020  Saarbrücken, Germany (Online)Talk: Statistical physics approaches to Unique Games (29 Jul)

4–5 Apr 2020POSTPONED Lake Michigan Workshop on Combinatorics and Graph Theory  Chicago, Illinois

19 Mar 2020POSTPONED Computer Science Theory Seminar, UIC  Chicago, IllinoisTitle: Statistical physics approaches to Unique Games (with Alex Kolla)

12 Feb 2020Discrete Mathematics Seminar, KdVI / CWI  Amsterdam, NetherlandsTitle: Statistical physics approaches to Unique Games

22 Jan 2020CS Theory Seminar, CU Boulder  Boulder, ColoradoTitle: Statistical physics approaches to Unique Games

25 Sep 2019CS Theory Seminar, CU Boulder  Boulder, ColoradoTitle: Graph colouring and the hardcore model

15 Jan – 17 May 2019Simons Institute: Geometry of Polynomials  Berkeley, CaliforniaTalk: Colouring locally sparse graphs via the hardcore model (17 Apr)

23–24 Aug 2018Algorithmic and Combinatorial Aspects of Partition Functions  Amsterdam, NetherlandsTalk: Fractional colouring of trianglefree graphs via the hardcore model (24 Aug)

9–10 May 2018Two oneday Colloquia in Combinatorics  London, UK

1 May 2018Applied Stochastics Seminar, Radboud University  Nijmegen, NetherlandsTitle: Using Gibbs measures in extremal combinatorics

26–30 Mar 2018Workshop on Graph limits  Janov, CzechiaTalk: Tight bounds on the coefficients of partition functions via stability (30 Mar)

19 Feb – 4 Mar 2018Graphs and Randomness, IMPA  Rio de Janeiro, Brazil

26 Jan 2018Discrete Mathematics Seminar, KdVI / CWI  Amsterdam, NetherlandsTitle: A new approach to Sidorenko's conjecture

18 Oct 2017Discrete Mathematics Seminar, KdVI / CWI  Amsterdam, NetherlandsTitle: Shamir's problem revisited

28 Aug – 1 Sep 2017EUROCOMB 2017  Vienna, AustriaTalk: Tight bounds on the coefficients of partition functions via stability (31 Aug, slides)

17–21 Jul 2017Novi Sad Workshop on Foundations Of Computer Science  Novi Sad, SerbiaTalk: Regularity inheritance in 3uniform hypergraphs (20 Jul)

11 May 2017Two oneday Colloquia in Combinatorics (invited)  London, UKTitle: Tight bounds on the coefficients of partition functions via stability

21 Feb 2017Combinatorics Seminar  Oxford, UKTitle: Extremal problems on colourings in cubic graphs via the Potts model

3 Feb 2017Discrete Mathematics Seminar, KdVI / CWI  Amsterdam, NetherlandsTitle: A probabilistic approach to bounding graph polynomials

27 Jan 2017PhD Seminar on Combinatorics, Games and Optimisation, LSE  London, UKTitle: A probabilistic approach to bounding graph polynomials

3 Aug 2016Student Combinatorics Day (invited)  Birmingham, UKTitle: On the average size of independent sets in trianglefree graphs (slides)

8 Apr 2016Seminário de Teoria da Computação, Combinatória e Otimização  São Paulo, BrazilTitle: Independent sets, matchings, and occupancy fractions II

3 Nov 2015Discrete Geometry and Combinatorics Seminar, UCL  London, UKTitle: Independent sets, matchings, and occupancy fractions

9 Oct 2015Mathematics Lunchtime Seminar, LSE  London, UKTitle: Independent sets, matchings, and occupancy fractions I

31 Aug – 4 Sep 2015EUROCOMB 2015  Bergen, NorwayTalk: Counting in hypergraphs via regularity inheritance (31 Aug, slides)

15 Apr 2015Postgraduate Combinatorial Conference, QMUL  London, UKTitle: Counting in hypergraphs via regularity inheritance

9 Apr 2015Combinatorics Seminar, Freie Universität  Berlin, GermanyTitle: Regularity inheritance in 3uniform hypergraphs

3 Jul 2014SUMMIT 190: Balogh, Csaba, Hajnal, and Pluhar are 190 (invited)  Szeged, HungaryTitle: Robustness of triangle factors

29 Nov 2013Mathematics Lunchtime Seminar, LSE  London, UKTitle: 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
Some demonstrations of my work can be found here
View my full CV as a pdf here
Positions held
 Postdoc at CU Boulder (2019Present)
 Research Fellow at the Simons Institute Geometry of Polynomials Program (2019)
 Postdoc at the University of Amsterdam (20172018)
Education
 Ph.D. Mathematics, London School of Economics and Political Science (20132017)
 M.Math, University of Cambridge (20122013)
 B.A. (Hons) Mathematics, University of Cambridge (20092012)
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