17th Jan: Julian Sahasrabudhe, Cambridge

A new lower bound for sphere packing

What is the maximum proportion of d-dimensional space that can be covered by disjoint, identical spheres? In this talk I will discuss a new lower bound for this problem, which is the first asymptotically growing improvement to Rogers' bound from 1947. Our proof is almost entirely combinatorial and reduces to a novel theorem about independent sets in graphs with bounded degrees and codegrees.

This is based on joint work with Marcelo Campos, Matthew Jenssen and Marcus Michelen.

24th Jan: Bobby Miraftab, Carleton University

Infinite Cycles in Cayley Graphs and Eigenvectors of Finite Support

This talk has two parts. In the first part, we explore infinite cycles in Cayley graphs. A weaker version of the celebrated Lovász conjecture asserts that every finite, connected Cayley graph contains a hamiltonian cycle. While infinite graphs cannot possess hamiltonian cycles in the traditional sense, there are natural analogues. We focus on a topological approach to this problem, demonstrating how some results for finite Cayley graphs can be extended to the infinite case. Additionally, we propose some open questions. In the second part, we turn our attention to constructions of locally finite graphs with eigenvectors of finite support, particularly for "very symmetric" graphs and over finite fields. We will provide some background, present basic constructions yielding "somewhat symmetric" examples, and discuss obstructions that preclude certain potential examples.

8th Nov: Matias Pavez-Signe, University of Chile

Ramsey numbers of cycles in random graphs

Let C_n denote the cycle on n vertices. We say a graph G is C_n-Ramsey if every 2-colouring of the edges of G contains a monochromatic copy of C_n. The classical Ramsey problem for cycles asks for determining the minimum number R(C_n) so that the complete graph on R(C_n) vertices is C_n-Ramsey. This talk will study when a random graph G(N,p) is C_n-Ramsey with high probability. In particular, we will show that even for very sparse edge probability p and N quite close to R(C_n), G(N,p) remains C_n-Ramsey.

15th Nov: Brett Kolesnik, University of Warwick

Graphical sequences and plane trees

We show that the asymptotic number of graphical sequences can be expressed in terms of Walkup’s formula for the number of plane trees. This yields a more detailed description of the asymptotics by Balister, Donderwinkel, Groenland, Johnston and Scott. Our proof is probabilistic, using what we call the Lévy–Khintchine method. We will discuss other applications of this method, and connections with additive number theory (subset counting formulas by von Sterneck and the Erdös–Ginzburg–Ziv theorem). Joint work with Michal Bassan (Oxford) and Serte Donderwinkel (Groningen).

22nd Nov: Maria Ivan, University of Cambridge

Euclidean Ramsey sets and the block sets conjecture

29th Nov: Peleg Michaeli, University of Oxford

Extremal and probabilistic aspects of graph rigidity

6th Dec: Debmalya Bandyopadhyay, University of Birmingham

Monochromatic tight cycle partitions in edge-coloured complete $k$-graphs

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph on $n$ vertices. A tight cycle is a $k$-uniform graph with its vertices cyclically ordered so that every~$k$ consecutive vertices form an edge, and any two consecutive edges share exactly~$k-1$ vertices. A result by Bustamante, Corsten, Frankl, Pokrovskiy and Skokan shows that all $r$-edge coloured $K_{n}^{(k)}$ can be partition into $c_{r,k}$ vertex disjoint monochromatic tight cycles. However, the constant $c_{r,k}$ is of tower-type. In this work, we show that $c_{r,k}$ is a polynomial in~$r$.