INTERSECTING FAMILIES WITH SUNFLOWER SHADOWS

A family F of k-subsets of {1,2, ..., n}is called t-intersecting if vertical bar F boolean AND F'vertical bar >= t for all F, F' is an element of F. A set E is called an r-sunflower shadow of F if one can choose r members F-1, F2, ..., F-r of F containing E and F-1 \ E, F-2 \ E, ..., F-r \ E are pairwise disjoint. Let D(n, k, t, l, r) = {D is an element of (([n])(k)) : vertical bar D boolean AND [t + (2r - 2)l]vertical bar >= t + (r - 1)l}. Motivated by our recent work [6] on intersecting families without unique shadow, we show that for l <= t, k >= t + (r - 1)l and n >= n(0)(k), D(n, k, t, l, r) is the only family attaining the maximum size among all t-intersecting families with all their lth shadows being r-sunflower.