On large intersecting subfamilies of uniform setfamilies

In an earlier work R. A. Duke and V. Rödl, The Erdős–Ko–Rado theorem for small families, J Combin Theory Ser A 65(2) (1994), 246–251 it was shown that for t a fixed positive integer and κ a real constant, 0 < κ < 1/2, if n is sufficiently large each family 𝒜 of ⌊κ n ⌋‐element subsets of [ n ] of size N (linear in n ) contains a t ‐intersecting subfamily of size at least (1 − o (1))κ N . Here we consider the case when t , the intersection size, is no longer bounded, specifically t = ⌊τ n ⌋ for 0 < τ < κ. We show that for sufficiently large n and N each family of this type contains an r ‐wise t ‐intersecting subfamily of size at least N 1−δ , and that, apart for the size of δ, this result is the best possible.