Ramsey Goodness of paths and unbalanced graphs

TL;DR

This paper extends the understanding of Ramsey goodness for paths, showing that under certain unbalanced conditions on the target graph, the path's Ramsey number matches the trivial lower bound for smaller path lengths than previously known.

Contribution

It proves that for unbalanced complete multipartite graphs, the path is Ramsey good at lengths proportional to the sum of the part sizes, improving earlier bounds.

Findings

Ramsey goodness holds for paths with respect to certain unbalanced graphs.

The path length threshold is reduced to approximately twice the total size of the target graph.

The result applies to multipartite graphs with specific unbalance conditions.

Abstract

Given graphs $G$ and $H$, we say that $G$ is $H$-$good$ if the Ramsey number $R(G,H)$ equals the trivial lower bound $(|G| - 1)(\chi(H) - 1) + \sigma(H)$, where $\chi(H)$ denotes the usual chromatic number of $H$, and $\sigma(H)$ denotes the minimum size of a color class in a $\chi(H)$-coloring of $H$. Pokrovskiy and Sudakov [Ramsey goodness of paths. Journal of Combinatorial Theory, Series B, 122:384-390, 2017.] proved that $P_n$ is $H$-good whenever $n\geq 4|H|$. In this paper, given $\varepsilon>0$, we show that if $H$ satisfy a special unbalance condition, then $P_n$ is $H$-good whenever $n \geq (2 + \varepsilon)|H|$. More specifically, we show that if $m_1,\ldots, m_k$ are such that $\varepsilon\cdot m_i \geq 2m_{i-1}^2$ for $2\leq i\leq k$, and $n \geq (2 + \varepsilon)(m_1 + \cdots + m_k)$, then $P_n$ is $K_{m_1,\ldots,m_k}$-good.