On July 14, 2020, Casey Tompkins from IBS Discrete Mathematics Group gave a talk on inverse Turán problems at the Discrete Math Seminar. The goal is for a fixed graph H and an integer k to find the minimum number of edges of a graph G such that that every subgraph of G with at least k edges has a subgraph isomorphic to H. The title of his talk is “Inverse Turán Problems“.

## Casey Tompkins, Inverse Turán Problems

For given graphs $G$ and $F$, the Turán number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Briggs and Cox introduced a dual version of this problem wherein for a given number $k$, one maximizes the number of edges in a host graph $G$ for which $ex(G,H) < k$. We resolve a problem of Briggs and Cox in the negative by showing that the inverse Turán number of $C_4$ is $\Theta(k^{3/2})$. More generally, we determine the order of magnitude of the inverse Turán number of $K_{s,t}$ for all $s$ and $t$. Addressing another problem of Briggs and Cox, we determine the asymptotic value of the inverse Turán number of the paths of length $4$ and $5$ and provide an improved lower bound for all paths of even length. We also obtain improved bounds on the inverse Turán number of even cycles

Joint work with Ervin Győri, Nika Salia and Oscar Zamora.

## Casey Tompkins gave a talk on the saturation problems in the context of Erdős-Szekeres theorems and posets at the Discrete Math Seminar

On April 14, 2020, Casey Tompkins from IBS discrete mathematics group gave a talk on the saturation version of the problems related to the Erdős-Szekeres theorem on convex k-gons, sequences, and posets. The title of his talk is “Saturation problems in the Ramsey theory of graphs, posets and point sets“.

## Casey Tompkins, Saturation problems in the Ramsey theory of graphs, posets and point sets

In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán’s classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets.

We also consider *semisaturation* problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erdős-Szekeres theorem on convex $k$-gons, as well as multiple semisaturation theorems for sequences and posets.

This project was joint work with Gábor Damásdi, Balázs Keszegh, David Malec, Zhiyu Wang and Oscar Zamora.

## Casey Tompkins gave a talk on the extremal problems for Berge hypergraphs at the discrete math seminar

On October 1, 2019, Casey Tompkins at IBS discrete mathematics group presented a seminar talk on an extension of Turán’s theorem to hypergraphs. The title of his talk was “Extremal problems for Berge hypergraphs“.

## Casey Tompkins, Extremal problems for Berge hypergraphs

Given a graph $G$, there are several natural hypergraph families one can define. Among the least restrictive is the family $BG$ of so-called Berge copies of the graph $G$. In this talk, we discuss Turán problems for families $BG$ in $r$-uniform hypergraphs for various graphs $G$. In particular, we are interested in general results in two settings: the case when $r$ is large and $G$ is any graph where this Turán number is shown to be eventually subquadratic, as well as the case when $G$ is a tree where several exact results can be obtained. The results in the first part are joint with Grósz and Methuku, and the second part with Győri, Salia and Zamora.

## Welcome Casey Tompkins, a new research fellow in the IBS discrete mathematics group

The IBS discrete mathematics group welcomes Dr. **Casey Tompkins**, a new research fellow at the IBS discrete mathematics group from September 1, 2019.

He received his Ph.D. from the Department of Mathematics at the Central European University in Budapest, Hungary in 2015 under the supervision of Prof. Gyula O. H. Katona.