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coloring of the positive integers

Source: 2024 CTST P9

March 11, 2024
combinatoricsRamsey Theory

Problem Statement

Color the positive integers by four colors c1,c2,c3,c4c_1,c_2,c_3,c_4. (1)Prove that there exists a positive integer nn and i,j∈{1,2,3,4}i,j\in\{1,2,3,4\},such that among all the positive divisors of nn, the number of divisors with color cic_i is at least greater than the number of divisors with color cjc_j by 33. (2)Prove that for any positive integer AA,there exists a positive integer nn and i,j∈{1,2,3,4}i,j\in\{1,2,3,4\},such that among all the positive divisors of nn, the number of divisors with color cic_i is at least greater than the number of divisors with color cjc_j by AA.
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