On the Trace of Finite Sets

Abstract For a family T of subsets of an n-set X we define the trace of it on a subset Y of X by T T (Y) = {F∩Y:Fϵ T }. We say that (m,n) → (r,s) if for every T with | T| ⩾m we can find a Y⊂X|Y| = s such that |T T (Y)| ⩾ r. We give a unified proof for results of Bollobas, Bondy, and Sauer concerning this arrow function, and we prove a conjecture of Bondy and Lovasz saying (⌋ n 2 4 ⌋ + n + 2,n)→ (3,7) , which generalizes Turan's theorem on the maximum number of edges in a graph not containing a triangle.