Families of Sets with Intersecting Clusters

William Y.C. Chen, Jiuqiang Liu and Larry X.W. Wang

  Abstract:  A family of k-subsets A1, A2,... , Ad on [n] = {1, 2,... , n} is called a (d, c)-cluster if the union A1A2 ∪...∪Ad 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- 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 satisfying 3 ≤ d k and n , if every (d, )-cluster is intersecting, then |F| ≤ with equality only when F is a complete star. This result is an extension of both Frankl's theorem and Mubayi's theorem.

  AMS Classification:  05D05

  Keywords:  clusters of subsets, Chvátal's simplex theorem, d-simplex, Erdös- Ko-Rado Theorem

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