A note on the size Ramsey number of powers of paths

TL;DR

This paper establishes new lower bounds on the size Ramsey number for powers of paths and related graphs, improving previous results and providing explicit bounds for large graphs.

Contribution

It proves novel lower bounds on size Ramsey numbers for powers of paths and connected graphs under specific conditions, extending known results.

Findings

Lower bounds for size Ramsey numbers of powers of paths are established.

Improved bounds for the size Ramsey number of three-color paths are provided.

Results depend on properties like average degree and independence number of the graph.

Abstract

Let $r\geq3$ be an integer such that $r-2$ is a prime power and let $H$ be a connected graph on $n$ vertices with average degree at least $d$ and $\alpha(H)\leq\beta n$, where $0<\beta<1$ is a constant. We prove that the size Ramsey number \[ \hat{R}({H};r) > \frac{{nd}}{2}{(r - 2)^2} - C\sqrt n \] for all sufficiently large $n$, where $C$ is a constant depending only on $r$ and $d$. In particular, for integers $k\ge1$, and $r\ge3$ such that $r-2$ is a prime power, we have that there exists a constant $C$ depending only on $r$ and $d$ such that $\hat{R}(P_{n}^{k}; r)> kn{(r - 2)^2}-C\sqrt n -\frac{{({k^2} + k)}}{2}{(r - 2)^2}$ for all sufficiently large $n$, where $P_{n}^{k}$ is the $kth$ power of $P_n$. We also prove that $\hat{R}(P_n,P_n,P_n)<764.1n$ for sufficiently large $n$. This result improves some results of Dudek and Pra{\l}at (\emph{SIAM J. Discrete Math.}, 31 (2017), 2079--2092 and \emph{Electron. J. Combin.}, 25 (2018), no.3, # P3.35).