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Familiy of finite subsets (sort of set theory)

Source: IMC 1997 day 1 problem 6

October 19, 2005
inductioncombinatorics solvedcombinatorics

Problem Statement

Suppose FF is a family of finite subsets of N\mathbb{N} and for any 2 sets A,B∈FA,B \in F we have A \cap B \not= \O. (a) Is it true that there is a finite subset YY of N\mathbb{N} such that for any A,B∈FA,B \in F we have A\cap B\cap Y \not= \O? (b) Is the above true if we assume that all members of FF have the same size?
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