Opuscula Math. 37, no. 4 (2017), 577-588
http://dx.doi.org/10.7494/OpMath.2017.37.4.577

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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