An explicit bound on the extremal function for complete graph minors

Abstract

Kostochka and Thomason independently proved that every graph with density at least $β\,t\sqrt{\log t}$ has a $K_t$ minor, for some constant $β$. Thomason later proved that the least such density is $(0.319\ldots+o(1))\,t\sqrt{\log t}$. However, the $o(1)$ term in this formula is not explicit, so the formula gives no bound for any specific value of $t$. The goal of this paper is a bound of the form $β\,t\sqrt{\log t}$ that holds for every $t$, with $β$ small. We prove that, for every integer $t\geqslant2$, every graph with density at least $4.5\,t\sqrt{\log t}$ has a $K_t$ minor.