Erdős Problem #610

The problem

For a graph $G$ let $\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (aside from isolated vertices). This is 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 $c>0$?

Statement quoted from erdosproblems.com/610, maintained by Thomas Bloom. Original source: [EGT92][Er94][Er99]. Current status there: proved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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