Linear dependencies among subsets of a finite set
Let integers \(k\) and \(t\) be given so that \(k>t\ge 0\). A family of \(k\)-sets \(F_1,\ldots,F_m\) with rational coefficients \(\alpha_1,\ldots,\alpha_m\) forms a \(0\)-design for \(k\) and \(t\) if, for any \(t\)-set \(G\), \(\sum \alpha_i=0\) (where the sum is taken over all \(i\) such that \(G\subset F_i)\). In this paper, the authors prove that, if \(\binom{t+1}{1}, \ldots, \binom{t+1}{k-t-1}\) have a common divisor \(d\) which does not divide \(\alpha_i\), then there exists an index \(j\) so that \(| F_i\cap F_j| = k-t-1\). Using this result, the authors deduce some other results in extremal set theory relating to problems posed by Erdős.
scientific article; zbMATH DE number 3843755