Opuscula Math. 37, no. 4 (2017), 577-588

Opuscula Mathematica

A general 2-part Erdȍs-Ko-Rado theorem

Gyula O. H. Katona

Abstract. A two-part extension of the famous Erdȍs-Ko-Rado Theorem is proved. The underlying set is partitioned into \(X_1\) and \(X_2\). Some positive integers \(k_i\), \(\ell_i\) (\(1\leq i\leq m\)) are given. We prove that if \(\mathcal{F}\)) is an intersecting family containing members \(F\) such that \(|F\cap X_1|=k_i\), \(|F\cap X_2|=\ell_i\) holds for one of the values \(i\) (\(1\leq i\leq m\)) then \(|\mathcal{F}|\) cannot exceed the size of the largest subfamily containing one element.

Keywords: extremal set theory, two-part problem, intersecting family.

Mathematics Subject Classification: 05D05.

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  • Gyula O. H. Katona
  • MTA Rényi Institute, Budapest Pf 127, 1364 Hungary
  • Communicated by Ingo Schiermeyer.
  • Received: 2016-12-31.
  • Revised: 2017-03-01.
  • Accepted: 2017-03-03.
  • Published online: 2017-04-28.
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Cite this article as:
Gyula O. H. Katona, A general 2-part Erdȍs-Ko-Rado theorem, Opuscula Math. 37, no. 4 (2017), 577-588, http://dx.doi.org/10.7494/OpMath.2017.37.4.577

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