Abstract:  A collection of k-subsets A₁, A₂,... , A_d on [n] = {1, 2,... , n}, not necessarily distinct, is called a (d, c)-cluster if the union A₁ ∪ A₂ ∪... âˆ?em> A_d contains at most ck elements with c < d. Let F be a family of k-subsets of an n-element set. We show that for k ≥ 2 and n ≥ k + 2, if every (k, 2)-cluster of F is intersecting, then F contains no (k âˆ?1)-dimensional simplices. This leads to an affirmative answer to Mubayi's conjecture for d = k based on Chvatal's simplex theorem. We also show that for any d with 3 ≤ d ≤ k and n ≥ , if every (d, d+1/2 )-cluster is intersecting, then |F| ≤ with equality only when F is a star. This result contains Frankl's theorem for d â‰?2 and Mubayi's theorem for d = 3 as special cases.

  AMS Classification:  05D05

  Keywords:  Clusters of subsets, Chvatal's simplex theorem, d-simplex, Erdös- Ko-Rado Theorem, Mubayi's conjecture

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