On the trace of finite sets

AbstractFor a family T of subsets of an n-set X we define the trace of it on a subset Y of X by TT(Y) = {F∩Y:FϵT}. We say that (m,n) → (r,s) if for every T with |T| ⩾m we can find a Y⊂X|Y| = s such that |TT(Y)| ⩾ r. We give a unified proof for results of Bollobàs, Bondy, and Sauer concerning this arrow function, and we prove a conjecture of Bondy and Lovász saying (⌋n24⌋ + n + 2,n)→ (3,7), which generalizes Turán's theorem on the maximum number of edges in a graph not containing a triangle