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# Seunghun Lee (이승훈), On colorings of hypergraphs embeddable in $\mathbb{R}^d$

## Monday, November 20, 2023 @ 4:30 PM - 5:30 PM KST

Given a hypergraph $H=(V,E)$, we say that $H$ is *(weakly) $m$-colorable* if there is a coloring $c:V\to [m]$ such that every hyperedge of $H$ is not monochromatic. The *(weak) chromatic number* of $H$, denoted by $\chi(H)$, is the smallest $m$ such that $H$ is $m$-colorable. A vertex subset $T \subseteq V$ is called a *transversal* of $H$ if for every hyperedge $e$ of $H$ we have $T\cap e \ne \emptyset$. The *transversal number* of $H$, denoted by $\tau(H)$, is the smallest size of a transversal in $H$. The *transversal ratio* of $H$ is the quantity $\tau(H)/|V|$ which is between 0 and 1. Since a lower bound on the transversal ratio of $H$ gives a lower bound on $\chi(H)$, these two quantities are closely related to each other.

Upon my previous presentation, which is based on the joint work with Joseph Briggs and Michael Gene Dobbins (https://www.youtube.com/watch?v=WLY-8smtlGQ), we update what is discovered in the meantime about transversals and colororings of geometric hypergraphs. In particular, we focus on chromatic numbers of $k$-uniform hypergraphs which are embeddable in $\mathbb{R}^d$ by varying $k$, $d$, and the notion of embeddability and present lower bound constructions. This result can also be regarded as an improvement upon the research program initiated by Heise, Panagiotou, Pikhurko, and Taraz, and the program by Lutz and Möller. We also present how this result is related to the previous results and open problems regarding transversal ratios. This presentation is based on the joint work with Eran Nevo.