On disjointly representable sets

Abstract

A system of setsE 1,E 2, ...,E kX is said to be disjointly representable if there existx 1,x 2, ...,x k teX such thatx i teE j i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems.

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Dedicated to Paul Erdős on his seventieth birthday

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Frankl, P., Pach, J. On disjointly representable sets. Combinatorica 4, 39–45 (1984). https://doi.org/10.1007/BF02579155

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 65
  • 05 A 05