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    A note on distinct differences in tt-intersecting families

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    For a family F\mathcal{F} of subsets of {1,2,,n}\{1,2,\ldots,n\}, let D(F)={FG:F,GF}\mathcal{D}(\mathcal{F}) = \{F\setminus G: F, G \in \mathcal{F}\} be the collection of all (setwise) differences of F\mathcal{F}. The family F\mathcal{F} is called a tt-intersecting family, if for some positive integer tt and any two members F,GFF, G \in \mathcal{F} we have FGt|F\cap G| \geq t. The family F\mathcal{F} is simply called intersecting if t=1t=1. Recently, Frankl proved an upper bound on the size of D(F)\mathcal{D}(\mathcal{F}) for the intersecting families F\mathcal{F}. In this note we extend the result of Frankl to tt-intersecting families

    Cross-intersecting families of vectors

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    Given a sequence of positive integers p=(p1,...,pn)p = (p_1, . . ., p_n), let SpS_p denote the family of all sequences of positive integers x=(x1,...,xn)x = (x_1,...,x_n) such that xipix_i \le p_i for all ii. Two families of sequences (or vectors), A,BSpA,B \subseteq S_p, are said to be rr-cross-intersecting if no matter how we select xAx \in A and yBy \in B, there are at least rr distinct indices ii such that xi=yix_i = y_i. We determine the maximum value of AB|A|\cdot|B| over all pairs of rr- cross-intersecting families and characterize the extremal pairs for r1r \ge 1, provided that minpi>r+1\min p_i >r+1. The case minpir+1\min p_i \le r+1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and answers a question of Zhang

    Regular Intersecting Families

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    We call a family of sets intersecting, if any two sets in the family intersect. In this paper we investigate intersecting families F\mathcal{F} of kk-element subsets of [n]:={1,,n},[n]:=\{1,\ldots, n\}, such that every element of [n][n] lies in the same (or approximately the same) number of members of F\mathcal{F}. In particular, we show that we can guarantee F=o((n1k1))|\mathcal{F}| = o({n-1\choose k-1}) if and only if k=o(n)k=o(n).Comment: 15 pages, accepted versio
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