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Combinatorics Seminar

Seminar organisers: Jan Grebík and Oleg Pikhurko

Warwick's combinatorics seminar in 2021-22 will be held in a hybrid format 2-3pm UK time on Wednesdays, occasionally 4-5pm UK time on Wednesdays (see the notes below), in the room B3.02 and on ZOOM.

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Meeting ID: 823 9801 5989
Passcode: WC2021

Term 1

Date Name Title Note
6 Oct Allan Sly (Princeton) Factor of IID for the Ising model on the tree online
20 Oct Henry Towsner (Pennsylvania) Regularity Lemmas as Structured Decompositions online
27 Oct Rob Silversmith (Warwick) Cross-ratios and perfect matchings  
3 Nov Joseph Hyde (Warwick) Progress on the Kohayakawa-Kreuter conjecture  
10 Nov Łukasz Grabowski (Lancaster) TBA  
17 Nov Natasha Morrison (Victoria) TBA online, 4pm
24 Nov      
1 Dec Martin Winter (Warwick) TBA  
8 Dec George Kontogeorgiou (Warwick) TBA  

Factor of IID for the Ising model on the tree (Allan Sly), 14:00, B3.02+ZOOM

It's known that there are factors of IID for the free Ising model on the d-regular tree when it has a unique Gibbs measure and not when reconstruction holds (when it is not extremal). We construct a factor of IID for the free Ising model on the d-regular tree in (part of) its intermediate regime, where there is non-uniqueness but still extremality. The construction is via the limit of a system of stochastic differential equations.

Regularity Lemmas as Structured Decompositions (Henry Towsner), 14:00, B3.02+ZOOM

One way of viewing Szemerédi's regularity lemma is that it gives a way of decomposing a graph (approximately) into a structured part (the "unary" data) and a random part. Then hypergraph regularity, the generalization to k-uniform hypergraphs, can be viewed as a decomposition into multiple "tiers" of structure - a unary part as well as a binary part and so on, and then finally a random part.

We'll discuss how an analytic approach can make these decompositions exact instances of the conditional expectation in probability, and how these analytic proofs relate to combinatorial proofs with explicit bounds. Finally, we'll discuss regularity lemmas for other mathematical objects, focusing on the example of ordered graphs and hypergraphs, and show how the "tiers of structures" perspective makes it possible to see regularity lemmas for other mathematical objects as examples of the regularity lemma for hypergraphs.

(No prior knowledge of the regularity lemma and its variants is assumed.)

Cross-ratios and perfect matchings (Rob Silversmith), 14:00, B3.02+ZOOM

I’ll describe a simple process from algebraic geometry that takes in a collection of 4-element subsets S1,S2,…,Sn-3 of [n], and outputs a nonnegative integer called a cross-ratio degree. I’ll discuss several interpretations of cross-ratio degrees in algebra, algebraic geometry, and tropical geometry, and present a combinatorial algorithm for computing them, due to C. Goldner. I’ll then present a perhaps-surprising upper bound for cross-ratio degrees in terms of matchings.

Progress on the Kohayakawa-Kreuter conjecture (Joseph Hyde), 14:00, B3.02+ZOOM

Let H1, ..., Hr be graphs. We write G(n,p) → (H1, ..., Hr) to denote the property that whenever we colour the edges of G(n,p) with colours from the set [r] := {1, ..., r} there exists some 1 ≤ i ≤ r and a copy of Hi in G(n,p) monochromatic in colour i.

There has been much interest in determining the asymptotic threshold function for this property. Rödl and Ruciński (1995) determined the threshold function for the general symmetric case; that is, when H1 = ... = Hr. A conjecture of Kohayakawa and Kreuter (1997), if true, would effectively resolve the asymmetric problem. Recently, the 1-statement of this conjecture was confirmed by Mousset, Nenadov and Samotij (2021+). The 0-statement however has only been proved for pairs of cycles, pairs of cliques and pairs of a clique and a cycle.


In this talk we introduce a reduction of the 0-statement of Kohayakawa and Kreuter's conjecture to a certain deterministic, natural subproblem. To demonstrate the potential of this approach, we show this subproblem can be resolved for almost all pairs of regular graphs (satisfying properties one can assume when proving the 0-statement).