The de Bruijn–Erdős theorem for hypergraphs

Abstract

Fix integers n ≥ r ≥ 2. A clique partition of \({{[n] \choose r}}\) is a collection of proper subsets \({A_1, A_2, \ldots, A_t \subset [n]}\) such that \({\bigcup_i{A_i \choose r}}\) is a partition of \({{[n]\choose r}}\) . Let cp(n, r) denote the minimum size of a clique partition of \({{[n] \choose r}}\) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3,$${\rm cp}(n, r) \geq (1 + o(1))n^{r/2} \quad \quad {\rm as} \, \, n \rightarrow \infty.$$We conjecture cp(n, r) = (1 + o(1))n r/2. This conjecture has already been verified (in a very strong sense) for r = 3 by Hartman–Mullin–Stinson. We give further evidence of this conjecture by constructing, for each r ≥ 4, a family of (1 + o(1))n r/2 subsets of [n] with the following property: no two r-sets of [n] are covered more than once and all but o(n r) of the r-sets of [n] are covered. We also give an absolute lower bound \({{\rm cp}(n, r) \geq {n \choose r}/{q + r - 1 \choose r}}\) when n = q 2 + q + r − 1, and for each r characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n, r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.