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Sets of consecutive integers

Source: Singapore MO 2011 senior round 2 Q4

June 25, 2011
combinatorics proposedcombinatorics

Problem Statement

Let nn and kk be positive integers with n≥k≥2n\geq k\geq 2. For i=1,…,ni=1,\dots,n, let SiS_i be a nonempty set of consecutive integers such that among any kk of them, there are two with nonempty intersection. Prove that there is a set XX of k−1k-1 integers such that each SiS_i, i=1,…,ni=1,\dots,n contains at least one integer in XX.
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