Jun Gao gave a talk on the maximum sum of the p-th power of degrees in a hypergraph without fixed subhypergraphs at the Discrete Math Seminar

On December 13, 2024, Jun Gao (高峻) from the IBS Extremal Combinatorics and Probability Group gave a talk at the Discrete Math Seminar on the maximum sum of the p-th power of degrees in a hypergraph without fixed subhypergraphs. The title of his talk is “Phase transition of degenerate Turán problems in p-norms“.

Jun Gao (高峻), Phase transition of degenerate Turán problems in p-norms

For a positive real number p, the p-norm Gp of a graph G is the sum of the p-th powers of all vertex degrees. We study the maximum p-norm exp(n,F) of F-free graphs on n vertices, focusing on the case where F is a bipartite graph. It is natural to conjecture that for every bipartite graph F, there exists a threshold pF such that for p<pF, the order of exp(n,F) is governed by pseudorandom constructions, while for p>pF, it is governed by star-like constructions. We determine the exact value of pF, under a mild assumption on the growth rate of ex(n,F). Our results extend to r-uniform hypergraphs as well.

We also prove a general upper bound that is tight up to a logn factor for exp(n,F) when p=pF.
We conjecture that this logn factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.

This is a joint work with Xizhi Liu, Jie Ma and Oleg Pikhurko.

Jun Gao gave a talk about the tight upper bound on the number of (k-1)-cliques in a k-critical graph at the Discrete Math Seminar

On August 30, 2022, Jun Gao (高峻) from the IBS Extremal Combinatorics and Probability Group gave a talk at the Discrete Math Seminar about the tight upper bound on the number of (k-1)-cliques in a k-critical graph, solving the conjecture of Abbott and Zhou in 1992. The title of his talk was “Number of (k-1)-cliques in k-critical graph“.

Jun Gao, Number of (k-1)-cliques in k-critical graph

We prove that for n>k3, if G is an n-vertex graph with chromatic number k but any its proper subgraph has smaller chromatic number, then G contains at most nk+3 copies of cliques of size k1. This answers a problem of Abbott and Zhou and provides a tight bound on a conjecture of Gallai.

This is joint work with Jie Ma.

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