# Large monochromatic components in 3-edge-colored Steiner triple systems

## TL;DR

This paper investigates the size of the largest monochromatic component in 3-edge-colored Steiner triple systems, showing that for most systems, this size is close to the total number of vertices, and relates it to the structure of 3-partite holes.

## Contribution

It proves that almost all Steiner triple systems have a monochromatic component of size nearly n in any 3-coloring, improving previous bounds and linking the size to the largest 3-partite hole.

## Key findings

- For almost all Steiner triple systems, the largest monochromatic component is (1-o(1))n.
- The lower bound depends on the size of the largest 3-partite hole.
- The bound is tight unless the coloring has a specific structure.

## Abstract

It is known that in any $r$-coloring of the edges of a complete $r$-uniform hypergraph, there exists a spanning monochromatic component. Given a Steiner triple system on $n$ vertices, what is the largest monochromatic component one can guarantee in an arbitrary 3-coloring of the edges? Gy\'arf\'as proved that $(2n+3)/3$ is an absolute lower bound and that this lower bound is best possible for infinitely many $n$. On the other hand, we prove that for almost all Steiner triple systems the lower bound is actually $(1-o(1))n$. We obtain this result as a consequence of a more general theorem which shows that the lower bound depends on the size of a largest \emph{3-partite hole} (that is, sets $X_1, X_2, X_3$ with $|X_1|=|X_2|=|X_3|$ such that no edge intersects all of $X_1, X_2, X_3$) in the Steiner triple system (Gy\'arf\'as previously observed that the upper bound depends on this parameter). Furthermore, we show that this lower bound is tight unless the coloring has a particular structure. We also suggest a variety of other Ramsey problems in the setting of Steiner triple systems.