Combinatorics Seminar

Combinatorics Seminar will be held 2-3pm on Fridays in B3.02, unless stated otherwise. In Term 1, the seminar is organised by Bertille Granet and Oleg Pikhurko.

Term 1

Date	Name	Title
9 Oct	No seminar (organisers away)
16 Oct	Joel Moreira (Warwick)	Growth rates for infinite sumsets in sets with positive density
23 Oct	Lu-Ming Zhang (LSE)	A Chvatal-type codegree condition for Hamiltonian cycles in k-uniform hypergraphs
30 Oct	Jane Tan (Oxford)	Hypergraphs with the smallest Hall ratio are still hard to colour
6 Nov	Arjun Ranganathan (UCL)
13 Nov	Lukas Michel (ETH)
20 Nov	Kohki Sakamoto (Tokyo)	Continuity of drift on lamplighter groups
27 Nov	Chiara Molinari (Warwick)
4 Dec	Antoine Poulin (Warwick)
11 Dec	Jasmin Katz (LSE)

16 Oct: Joel Moreira

Growth rates for infinite sumsets in sets with positive density

In 2019, in joint work with Florian Richter and Donald Robertson, we proved that any set A of natural numbers with positive upper density contains a sumset B+C where B and C are both infinite sets of natural numbers. An example of Bernard Host shows that, in general, B and C may be forced to have zero density.

There are now three or four proofs of the sumset theorem, but none of them give any growth condition on the sizes of B and C beyond Host's example. I will present ongoing joint work with Leo Verteegen, where we explore this problem. It turns out that the answer changes drastically, depending on the exact formulation of the question asked.

23 Oct: Lu-Ming Zhang

A Chvatal-type codegree condition for Hamiltonian cycles in k-uniform hypergraphs

We prove an asymptotic Chvatal-type codegree criterion for tight Hamiltonian cycles in $k$-uniform hypergraphs for every fixed $k\ge3$. The criterion allows small degrees to be compensated by large degrees in the link graphs of $(k-2)$-tuples. It strengthens the celebrated asymptotic Dirac-type theorem by Rodl, Rucinski and Szemeredi and implies, as a corollary, a Posa-type criterion conjectured by Schulke.

30 Oct: Jane Tan

Hypergraphs with the smallest Hall ratio are still hard to colour

For a hypergraph $H$, the Hall ratio ($\rho(H)$), fractional chromatic number ($\chi_f(H)$), and chromatic number ($\chi(H)$) satisfy the chain of inequalities $\rho(H)\leq \chi_f(H)\leq \chi(H)$. A natural question with a long history is whether each parameter can be bounded in terms of the preceding ones. We will survey what is known about the gaps between these parameters, and the interesting behaviour that occurs when the Hall ratio is as small as possible. This is based on recent joint work with Veronica Bitonti, Freddie Illingworth, and Pedram Noohi.

20 Nov: Kohki Sakamoto

Continuity of drift on lamplighter groups
Lamplighter groups are classical examples in geometric group theory, with a simple interpretation in terms of a lamplighter walking around a graph and switching lamps on and off. Under this interpretation, the standard word metrics on these groups can be described in terms of the travelling salesman problem on the graph. In this talk, we will consider random walks on lamplighter groups over Z^d with d≧3, and show that the asymptotic speed is continuous as a function of the driving measure. This is joint work with Eduardo Silva.