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Set: Prove that intersection of all A_i's is not a null set

Source: India tst 2006 p12

June 27, 2012
inductioncombinatorics unsolvedcombinatorics

Problem Statement

Let A1,A2,…,AnA_1,A_2,\ldots,A_n be subsets of a finite set SS such that ∣Aj∣=8|A_j|=8 for each jj. For a subset BB of SS let F(B)={j∣1≤j≤n  and Aj⊂B}F(B)=\{j \mid 1\le j\le n \ \ \text{and} \ A_j \subset B\}. Suppose for each subset BB of SS at least one of the following conditions holds
(a) ∣B∣>25|B| > 25,
(b) F(B)={\O},
(c) \bigcap_{j\in F(B)} A_j \neq {\O}.
Prove that A_1\cap A_2 \cap \cdots \cap A_n \neq {\O}.
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