On the coloring of 3-element subsets

TL;DR

This paper proves a specific coloring property for 3-element subsets of certain integers, showing that for numbers of the form 8k+1 with 8k-1 prime, a coloring exists where sets sharing two elements are differently colored.

Contribution

It establishes the existence of a special coloring scheme for 3-element subsets under particular number-theoretic conditions, a novel result in combinatorial coloring.

Findings

Existence of a coloring for 3-element subsets when n=8k+1 and 8k-1 is prime.

Sets sharing exactly two elements have different colors under this coloring.

The result links number theory with combinatorial coloring problems.

Abstract

In this paper we prove that for numbers $n = 8k+1$, such that $8k-1$ is prime, there exists the coloring of all 3-element subsets of {1, ..., n} such that any sets $A, B: |A\cap B|=2$ have different colors.