Intersecting balanced families of sets
Let \(N\) be an \(n\)-element set and \({\mathcal A}\subseteq 2^N\). For \(A\in \mathcal A\) denote by \(\chi(A)\) the characteristic vector of \(A\). The family \(\mathcal A\) is called balanced, if \(\sum_{A\in\mathcal A} \chi(A)=(1,1,\dots,1)\). The authors study balanced families in which the intersection of any \(t\geq 2\) sets consists of at least \(k\geq 1\) elements. The main result is that the maximum size of a set \(A\) in \(\mathcal A\) is at least \(k^{1/t}b^{1-1/t}\). This bound follows from counting arguments. An algebraic construction is provided that shows that this lower bound is asymptotically sharp.
scientific article; zbMATH DE number 1587279
scientific article