# 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**.

Join Zoom Meeting

https://us02web.zoom.us/j/82398015989?pwd=MVh0MiswazlzajVrRk9BQ3laK1E0Zz09

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 S_{1},S_{2},…,S_{n-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**

_{1}, ..., H

_{r}be graphs. We write G(n,p) → (H

_{1}, ..., H

_{r}) 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 H

_{i}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 H_{1} = ... = H_{r}. 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).