Discrete Mathematics 2010-2022

A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $\min\{c, |e|\}$ distinct colours. Let $\chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $\chi(t,c)$ is finite: they showed that $\chi(t,c)$ is finite whenever $t \geq c$ and unbounded when $t\leq c-2$. The boundary case $\chi(c-1, c)$ has remained elusive for some time: $\chi(1,2)$ is known to be finite by an easy classical result, and $\chi(2,3)$ was shown to be finite by Chung and independently by Colucci and Gy\'{a}rf\'{a}s in 2013. In this talk, we present some recent work with Kevin Hendrey, Freddie Illingworth and Nina Kam\v{c}ev in which we fill in this gap by showing that $\chi(c-1, c)$ is finite in general.

Meeting link:
\[
\text{https://tugraz.webex.com/tugraz/j.php?MTID=m8500c46344212abf0fa37925da5ef9bf}
\]