Cross-intersecting sub-families of hereditary families |
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Authors: | Peter Borg |
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Affiliation: | Department of Mathematics, University of Malta, Msida MSD 2080, Malta |
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Abstract: | Families A1,A2,…,Ak of sets are said to be cross-intersecting if for any i and j in {1,2,…,k} with i≠j, any set in Ai intersects any set in Aj. For a finite set X, let X2 denote the power set of X (the family of all subsets of X). A family H is said to be hereditary if all subsets of any set in H are in H; so H is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family H≠{∅} of X2 and any k?|X|+1, both the sum and the product of sizes of k cross-intersecting sub-families A1,A2,…,Ak (not necessarily distinct or non-empty) of H are maxima if A1=A2=?=Ak=S for some largest starSofH (a sub-family of H whose sets have a common element). We prove this for the case when H is compressed with respect to an element x of X, and for this purpose we establish new properties of the usual compression operation. As we will show, for the sum, the condition k?|X|+1 is sharp. However, for the product, we actually conjecture that the configuration A1=A2=?=Ak=S is optimal for any hereditary H and any k?2, and we prove this for a special case. |
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Keywords: | Chvá tal?s conjecture Cross-intersecting families Extremal set theory Hereditary families Intersecting families |
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