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intersection of 2^{n-1} subsets of F is not empty

Source: Ukraine TST 2009 p3

May 3, 2020

SubsetsSetscombinatorics

Problem Statement

Let

S

be a set consisting of

n

elements,

F

a set of subsets of

S

consisting of

2^{n-1}

subsets such that every three such subsets have a non-empty intersection. a) Show that the intersection of all subsets of

F

is not empty. b) If you replace the number of sets from

2^{n-1}

with

2^{n-1}-1

, will the previous answer change?