Erdős Problem #610

declared status 'proved'. Formalized: no. For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number). Estimate $\tau(G)$. In particular, is it true that if $G$ has $n$ vertices then$$\tau(G) \leq n-\omega(n)\sqrt{n}$$for some $\omega(n)\to \infty$, or even$$\tau(G) \leq n-c\sqrt{n\log n}$$for some absolute constant $c>0$? Current best: A problem of Erd\H{o}s, Gallai, and Tuza [EGT92], who proved that$$\tau(G) \leq n-\sqrt{2n}+O(1).$$This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\sqrt{n\log n})$, which follows from the lower bound for $R(3,k)$ by Kim [Ki95] (see [165]). Prize: no. Tags: graph theory.

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