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6-coloring [n], set x+y+z=0 mod n and x,y,z have the same/different color

Source: VJIMC 2008 1.4

June 17, 2021
combinatoricsinequalitiescombinatorial inequalityColoringnumber theory

Problem Statement

The numbers of the set {1,2,…,n}\{1,2,\ldots,n\} are colored with 66 colors. Let S:={(x,y,z)∈{1,2,…,n}3:x+y+z≡0(modn) and x,y,z have the same color}S:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have the same color}\}and D:={(x,y,z)∈{1,2,…,n}3:x+y+z≡0(modn) and x,y,z have three different colors}.D:=\{(x,y,z)\in\{1,2,\ldots,n\}^3:x+y+z\equiv0\pmod n\text{ and }x,y,z\text{ have three different colors}\}.Prove that ∣D∣≤2∣S∣+n22.|D|\le2|S|+\frac{n^2}2.
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