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In this lecture, Professor Jozef Skokan from the London School of Economics and Political Science explores the connection between the chromatic threshold problem in extremal combinatorics and the (p,q)-theorem in discrete geometry. Discover how graphs with bounded clique number and specific density conditions lead to a (p,q)-theorem for the dual of maximal independent sets hypergraph. Learn about results that strengthen previous work by Thomassen and Nikiforov on the chromatic threshold of cliques. Explore how these graphs demonstrate "bounded complexity" as blow-ups of constant size graphs with identical clique numbers, which enhances findings by Luczak, Goddard-Lyle on homomorphism threshold of cliques and improves upon Oberkampf and Schacht's quantitative results. Understand why the decisive factor in such blow-up phenomena is the density condition on co-neighbourhoods rather than the minimum degree condition typically considered in literature. This talk presents joint research with Hong Liu and Chong S, delivered by Skokan, whose research interests include extremal and probabilistic combinatorics and their connections to other mathematical areas.
