Ramsey's theorem states that, for a fixed graph , every 2-edge-colouring of contains a monochromatic copy of whenever is large enough. Perhaps one of the most natural questions after Ramsey's theorem is then how many copies of monochromatic can be guaranteed to exist. To formalise this question, let the Ramsey multiplicity …
There has been much research on finding a large rainbow matching in a properly edge-colored graph, where a proper edge coloring is a coloring of the edge set such that no same-colored edges are incident. Barát, Gyárfás, and Sárközy conjectured that in every proper edge coloring of a multigraph (with parallel edges allowed, but not …
A graph is common if the number of monochromatic copies of in a 2-edge-colouring of the complete graph is minimised by the random colouring. Burr and Rosta, extending a famous conjecture by Erdős, conjectured that every graph is common. The conjectures by Erdős and by Burr and Rosta were disproved by Thomason and …
The study of Hamiltonian graphs, i.e. finite graphs having a cycle that contains all vertices of the graph, is a central theme of finite graph theory. For infinite graphs such a definition cannot work, since cycles are finite. We shall debate possible concepts of Hamiltonicity for infinite graphs and eventually follow the topological approach by …
I will go over the history on the study of the set of cycle lengths of graphs with large average degree or chromatic number, and discuss recent work with Richard Montgomery on this topic. In particular, we will see the divergence of harmonic sum of odd cycle lengths in graphs with large chromatic number and …
In this talk I will state a generalisation of the even directed cycle problem, which asks whether a given digraph contains a directed cycle of even length, to orientations of regular matroids. Motivated by this problem, I will define non-even oriented matroids generalising non-even digraphs, which played a central role in resolving the computational complexity of …
In combinatorics, Hopf algebras appear naturally when studying various classes of combinatorial objects, such as graphs, matroids, posets or symmetric functions. Given such a class of combinatorial objects, basic information on these objects regarding assembly and disassembly operations are encoded in the algebraic structure of a Hopf algebra. One then hopes to use algebraic identities of …
Given a graph G=(V,E), the independence complex of G is the abstract simplicial complex I(G) on V whose faces are the independent sets of G. A graph is ternary if it does not contain an induced cycle of length divisible by three. Kalai and Meshulam conjectured that if G is ternary then the sum of the Betti numbers …
In an n-vertex graph, there must be a clique or stable set of size at least , and there are graphs where this bound is attained. But if we look at graphs not containing a fixed graph H as an induced subgraph, the largest clique or stable set is bigger. Erdős and Hajnal conjectured in 1977 that …
