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A_1 \cup A_2 \cup ... \cup A_n may be colored with 2 colors

Source: Indian Postal Coaching 2007 set 6 p4

May 25, 2020
ColoringcombinatoricsSubsets

Problem Statement

Let A1,A2,...,AnA_1,A_2,...,A_n be nn finite subsets of a set X,n≥2X, n \ge 2, such that (i) ∣Ai∣≥2,1≤i≤n|A_i| \ge 2, 1 \le i \le n, (ii) ∣Ai∩Aj∣≠1,j≤i<j≤n |A_i \cap A_j | \ne 1, j \le i < j \le n. Prove that the elements of A1∪A2∪...∪AnA_1 \cup A_2 \cup ... \cup A_n may be colored with 22 colors so that no AiA_i is colored by the same color.
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