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$(A − A) \cap (B − B)$ is nonempty

Source: 11-th Hungary-Israel Binational Mathematical Competition 2000

April 22, 2007
functionpigeonhole principlecombinatorics proposedcombinatorics

Problem Statement

Let AA and BB be two subsets of S={1,2,...,2000}S = \{1, 2, . . . , 2000\} with ∣A∣⋅∣B∣≥3999|A| \cdot |B| \geq 3999. For a set XX , let X−XX-X denotes the set {s−t∣s,t∈X,s≠t}\{s-t | s, t \in X, s \not = t\}. Prove that (A−A)∩(B−B)(A-A) \cap (B-B) is nonempty.
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