# Restricted online Ramsey numbers of matchings and trees

## TL;DR

This paper investigates the restricted online Ramsey numbers for matchings and trees, demonstrating bounds on the number of edges Builder needs to reveal to find large monochromatic structures in multi-colorings of complete graphs.

## Contribution

It establishes new bounds on the number of revealed edges required for Builder to locate large monochromatic matchings and trees in multi-colorings.

## Key findings

- Builder can find a monochromatic matching of size at least (n - t + 1)/(t + 1) with O(n log t) edges revealed.
- In 3-colorings, Builder can locate a monochromatic tree on at least n/2 vertices with at most 5n edges revealed.

## Abstract

Consider a two-player game between players Builder and Painter. Painter begins the game by picking a coloring of the edges of $K_n$, which is hidden from Builder. In each round, Builder points to an edge and Painter reveals its color. Builder's goal is to locate a particular monochromatic structure in Painter's coloring by revealing the color of as few edges as possible. The fewest number of turns required for Builder to win this game is known as the restricted online Ramsey number. In this paper, we consider the situation where this "particular monochromatic structure" is a large matching or a large tree. We show that in any $t$-coloring of $E(K_n)$, Builder can locate a monochromatic matching on at least ${n-t+1\over t+1}$ edges by revealing at most $O(n\log t)$ edges. We show also that in any $3$-coloring of $E(K_n)$, Builder can locate a monochromatic tree on at least $n/2$ vertices by revealing at most $5n$ edges.