Testing Intersectingness of Uniform Families

A set family ${\cal F}$ is called intersecting if every two members of ${\cal F}$ intersect, and it is called uniform if all members of ${\cal F}$ share a common size. A uniform family ${\cal F} \subseteq \binom{[n]}{k}$ of $k$-subsets of $[n]$ is $\varepsilon$-far from intersecting if one has to remove more than $\varepsilon \cdot \binom{n}{k}$ of the sets of ${\cal F}$ to make it intersecting. We study the property testing problem that given query access to a uniform family ${\cal F} \subseteq \binom{[n]}{k}$, asks to distinguish between the case that ${\cal F}$ is intersecting and the case that it is $\varepsilon$-far from intersecting. We prove that for every fixed integer $r$, the problem admits a non-adaptive two-sided error tester with query complexity $O(\frac{\ln n}{\varepsilon})$ for $\varepsilon \geq \Omega( (\frac{k}{n})^r)$ and a non-adaptive one-sided error tester with query complexity $O(\frac{\ln k}{\varepsilon})$ for $\varepsilon \geq \Omega( (\frac{k^2}{n})^r)$. The query complexities are optimal up to the logarithmic terms. For $\varepsilon \geq \Omega( (\frac{k^2}{n})^2)$, we further provide a non-adaptive one-sided error tester with optimal query complexity of $O(\frac{1}{\varepsilon})$. Our findings show that the query complexity of the problem behaves differently from that of testing intersectingness of non-uniform families, studied recently by Chen, De, Li, Nadimpalli, and Servedio (ITCS, 2024).