On the Erdős–Ko–Rado theorem and the Bollobás theorem for tt-intersecting families

A family FF is tt-intersecting   if any two members have at least tt common elements. Erdős, Ko and Rado (1961) proved that the maximum size of a tt-intersecting family of subsets of size kk is equal to n−tk−t if n≥n0(k,t)n≥n0(k,t). Alon, Aydinian and Huang (2014) considered families generalizing intersecting families, and proved the same bound. In this paper, we give a strengthening of their result by considering families generalizing tt-intersecting families for all t≥1t≥1. In 2004, Talbot generalized Bollobás’s Two Families Theorem (Bollobás, 1965) to tt-intersecting families. In this paper, we proved a slight generalization of Talbot’s result by using the probabilistic method.