Smallest Set-Transversals of k-Partitions

Abstract

We asymptotically solve an open problem raised independently by Sterboul (Colloq Math Soc J Bolyai 3:1387–1404, 1973), Arocha et al. (J Graph Theory 16:319–326, 1992) and Voloshin (Australas J Combin 11:25–45, 1995). For integers n ≥ k ≥ 2, let f(n, k) denote the minimum cardinality of a family ${\mathcal H}$ of k-element sets over an n-element underlying set X such that every partition ${X_1\cup\cdots\cup X_k=X}$ into k nonempty classes completely partitions some ${H\in\mathcal H}$;  that is, ${|H\cap X_i|=1}$ holds for all 1 ≤ i ≤ k. This very natural function—whose defining property for k = 2 just means that ${\mathcal H}$ is a connected graph—turns out to be related to several extensively studied areas in combinatorics and graph theory. We prove general estimates from which ${ f(n,k) = (1+o(1))\, \tfrac{2}{n}\,{n\choose k}}$ follows for every fixed k, and also for all k = o(n ^1/3), as n → ∞. Further, we disprove a conjecture of Arocha et al. (1992). The exact determination of f(n,k) for all n and k appears to be far beyond reach to our present knowledge, since e.g. the equality ${f(n,n-2)={n-2\choose 2}-{\rm ex}(n,\{C_3,C_4\})}$ holds, where the last term is the Turán number for graphs of girth 5.