Intersecting families with large covering number - On a conjecture of Frankl, Ota, and Tokushige

Abstract

Stability theorems on intersecting families are often formulated via the covering number of set families. Let $M(n,k,t)$ denote the maximum size of a $k$-uniform intersecting family $\mathcal F\subseteq \binom{[n]}{k}$ with covering number $t$. It is known that for any fixed $t$ and fixed $k$ sufficiently large relative to $t$, one has $M(n,k,t)=(p_{t-1}(k)+o(1))(k)\binom{n}{k}$ as $n$ tends to infinity, where $p_{t-1}(k)$ is the maximum number of covers of size $t-1$ that a $k$-uniform intersecting family can have provided its covering number is $t-1$. Frankl, Ota, and Tokushige conjectured that $p_t(k)=k^t-\binom{t}{2}k^{t-1}+O_t(k^{t-2})$ for fixed $t$ as $k$ tends to infinity and they also showed that $p_t(k)=k^t-b(t)k^{t-1}+O_t(k^{t-2})$, where $b(t)$ is defined as the extremum of the following minimization problem. Suppose a $k$-uniform intersecting family $\mathcal F$ of covering number $t$ can be partitioned into $\mathcal F_1,\dots,\mathcal F_t$ such that for every $i$ there exists a set $X_i$ with $X_i \subseteq F_i$ for all $F_i\in\mathcal F_i$ and the $X_i$'s are pairwise disjoint and non-empty. Then $b(t)$ is the minimum of $\sum_{i=1}^t(k-|X_i|)$ over all such families, partitions and $X_i$'s. In accordance with the conjecture on $p_t(k)$, Frankl, Ota, and Tokushige conjectured $b(t)=\binom{t}{2}$ for all $t\ge 2$ and proved $\frac{1}{\sqrt{2}}\lfloor \frac{t-1}{2}\rfloor^{3/2}\le b(t)\le \binom{t}{2}$. In this note, we present a counterexample to the conjecture showing $b(t)\le 2\sqrt2\,t^{3/2}\sqrt{\log t}$ $+O(t^{3/2})$ and improve the lower bound to $\frac{2}{3^{3/4}}t^{3/2}-O(t)\le b(t)$. The counterexample was found by ChatGPT 6.1Sol, the lower bound proof is due to the human author.