Speaker: Filip Kučerák telling us about Sidorenko property and forcing in regular tournaments
Abstract: We say that a graph has the Sidorenko property if it holds that the asymptotic minimizer of the number of copies of H among graphs with edge density p is a random graph, i.e., a graph where we include each edge independently with probability p.
A famous conjecture of Sidorenko and Erdős-Simonovits states that a graph has the Sidorenko property if and only if it is bipartite. The claim has been verified for various graph classes, such as even cycles, complete graphs, or trees; however, a general resolution seems out of reach.
In this talk, we survey the Sidorenko property in the setting of tournaments. In contrast to the graph setting, a complete characterization of tournaments that have the so-called tournament Sidorenko property is known. Noel, Ranganathan and Simbaqueba have shown, in their work on quasirandomness, that by requiring the random tournament to be the asymptotic minimizer only among the regular tournaments, we arrive at a weaker property. In their work, they present a finite set of tournaments satisfying only this weaker notion and leave the classification of such tournaments as an open problem. In joint work with Kráľ, Krnc, Lidický and Volec, we give a complete characterization of this set.