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Uniform s-Cross-Intersecting Families

  • PETER FRANKL (a1) and ANDREY KUPAVSKII (a2) (a3)

In this paper we study a question related to the classical Erdős–Ko–Rado theorem, which states that any family of k-element subsets of the set [n] = {1,. . .,n} in which any two sets intersect has cardinality at most $\binom{n-1}{k-1}$ .

We say that two non-empty families ${\mathcal A}, {\mathcal B}\subset \binom{[n]}{k}$ are s-cross-intersecting if, for any A ${\mathcal A}$ , B ${\mathcal B}$ , we have |AB| ≥ s. In this paper we determine the maximum of | ${\mathcal A}$ |+| ${\mathcal B}$ | for all n. This generalizes a result of Hilton and Milner, who determined the maximum of | ${\mathcal A}$ |+| ${\mathcal B}$ | for non-empty 1-cross-intersecting families.

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[1] Erdős, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets. Quart. J. Math. 12 313320.
[2] Frankl, P. and Kupavskii, A. A size-sensitive inequality for cross-intersecting families, to appear in European Journal of Combinatorics, arXiv:1603.00936
[3] Frankl, P. and Tokushige, N. (1992) Some best possible inequalities concerning cross-intersecting families. J. Combin. Theory Ser. A 61 8797.
[4] Hilton, A. J. W. and Milner, E. C. (1967) Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford 18 369384.
[5] Lau, L. C., Ravi, R. and Singh, M. (2011) Iterative Methods in Combinatorial Optimization, Cambridge University Press.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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