This week's seminar is given by our own Victor Falgas Ravry.
Title: The Mubayi—Terry problem for locally sparse multigraphs
Abstract: A multigraph is (s,q)-sparse if every s-set of vertices in G supports at most q edges, counting multiplicities. How large can the product of the edge multiplicities be in an (s,q)-sparse multigraph on n vertices?
This question was posed by Mubayi and Terry in 2016, with motivation coming from hypergraph container theory. It can also be seen as a natural attempt to generalise classical results from extremal graph theory to the setting of multigraphs with bounded edge multiplicities. That one seeks to maximise a product rather than a sum results in some unusual features, such as extremal constructions with parts containing an asymptotically transcendental proportion of the vertices.
Despite some recent progress, the answer to Mubayi and Terry's question is still poorly understood for general (s,q). In this talk, I will give some background and motivation for the study of (s,q)-sparse multigraphs, before going over the (few) know results and (many) open problems in the area.
