Thresholds for the Frankl-Wang $3/7$ conjecture on maximum-degree ratios

Abstract

Let $\mathcal{F}\subset \left(\genfrac{}{}{0ex}{}{[n]}{k}\right)$ be an intersecting family, $\hat{I}”(\mathcal{F})={\max }_{x\in [n]}|\{F\in \mathcal{F}:x\in F\}|$, and $\varrho (\mathcal{F})=\hat{I}”(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$ and $|\mathcal{F}|>\left(\genfrac{}{}{0ex}{}{n-3}{k-3}\right)$, then $\varrho (\mathcal{F})\ge 3/7$; the constant $3/7$ is sharp because of the Fano-plane construction. In this note we obtain three results. First, we show that no linear threshold $n>Ck$ can be sufficient: using a truncated Fano-plane construction we exhibit, for every constant $C$ and all large $k$, an intersecting family with $n>Ck$, $|\mathcal{F}|>\left(\genfrac{}{}{0ex}{}{n-3}{k-3}\right)$, yet $\varrho (\mathcal{F})<3/7$. In particular, the original condition $n>100k$ does not guarantee the conclusion. Second, for $k=3$ we prove that $\varrho (\mathcal{F})\ge 3/7$ holds for every nonempty intersecting $3$-uniform family; the proof is nontrivial and does not rely on any assumption on $n$ or $|\mathcal{F}|$. Third, using the classical pseudo-sunflower bound $|\mathcal{F}|\le t^k$ (for families containing no pseudo-sunflower of size $t+1$), we obtain a completely explicit polynomial threshold for all $k\ge 4$: if $n>(k-3)(7k^4+k)+3$ and $|\mathcal{F}|>\left(\genfrac{}{}{0ex}{}{n-3}{k-3}\right)$, then $\varrho (\mathcal{F})\ge 3/7$. In particular, the simplified bound $n>7k^5$ is sufficient for every $k\ge 4$.

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