Monochromatic loose path partitions in k-uniform hypergraphs

TL;DR

This paper proves a conjecture about monochromatic loose path partitions in 2-colored complete k-uniform hypergraphs, confirming it for all k ≥ 3 and improving previous bounds on uncovered vertices.

Contribution

It establishes the conjecture for all k ≥ 3, advancing understanding of monochromatic path partitions in hypergraphs.

Findings

Confirmed the conjecture for all k ≥ 3.

Improved bounds on the number of uncovered vertices.

Extended previous results from specific cases to general k.

Abstract

A conjecture of Gy\'{a}rf\'{a}s and S\'{a}rk\"{o}zy says that in every $2$-coloring of the edges of the complete $k$-uniform hypergraph $K_n^k$, there are two disjoint monochromatic loose paths of distinct colors such that they cover all but at most $k-2$ vertices. A weaker form of this conjecture with $2k-5$ uncovered vertices instead of $k-2$ is proved, thus the conjecture holds for $k=3$. The main result of this paper states that the conjecture is true for all $k\ge 3$.