A set family \(\mathcal {F}\) is called intersecting if every two members of \(\mathcal {F}\) intersect, and it is called uniform if all members of \(\mathcal {F}\) share a common size. A uniform family \(\mathcal {F}\subseteq \left( {\begin{array}{c}[n]\\ k\end{array}}\right)\) of k-subsets of [n] is \(\varepsilon\) -far from intersecting if one has to remove more than \(\varepsilon \cdot \left( {\begin{array}{c}n\\ k\end{array}}\right)\) of the sets of \(\mathcal {F}\) to make it intersecting. We study the property testing problem that given query access to a uniform family \(\mathcal {F}\subseteq \left( {\begin{array}{c}[n]\\ k\end{array}}\right)\) , asks to distinguish between the case that \(\mathcal {F}\) 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 \(\varepsilon \ge \Omega ( (\frac{k}{n})^r)\) and a non-adaptive one-sided error tester with query complexity \(O(\frac{\ln k}{\varepsilon })\) for \(\varepsilon \ge \Omega ( (\frac{k^2}{n})^r)\) . The query complexities are optimal up to the logarithmic terms. For \(\varepsilon \ge \Omega ( (\frac{k^2}{n})^2)\) , 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 et al. (2024).
