MathLabs

Problem 3

Let S={1,2,3,…,280}S=\{1,2,3,\ldots,280\}. Find the smallest integer n such that each n-element subset of S contains five numbers which are pairwise relatively prime.
Step 4 of 5: Add five explicit coprime blocks
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A1,A2,A3 each have 6 elements; B1,B2 each have 5 elementsA_1,A_2,A_3\text{ each have 6 elements; }B_1,B_2\text{ each have 5 elements}
Detailed analysis

Use the five disjoint blocks A1={2·41,3·37,5·31,7·29,11·23,13·19}, A2={2·37,3·31,5·29,7·23,11·19,13·17}, A3={2·31,3·29,5·23,7·19,11·17,13·13}, B1={2·29,3·23,5·19,7·17,11·13}, and B2={2·23,3·19,5·17,7·13,11·11}. Each block is pairwise coprime, all entries lie below 280, and they are disjoint from P and one another. Thus their union with P has 60+6+6+6+5+5=88 elements.