# How Ramsey theory can be used to solve Harary's problem for $K_{2,k}$

## TL;DR

This paper explores how Ramsey theory can be applied to determine upper bounds for Ramsey numbers involving the bipartite graph $K_{2,k}$, extending previous results and providing sharp bounds for specific graph classes.

## Contribution

It introduces new bounds for $r(K_{2,k},G)$ and generalizes Harary's problem for these graphs using Ramsey theory techniques.

## Key findings

- Established that $r(C_4,G) \,\leq\, kq+1$ for certain graphs G.
- Proved that equality holds when $G \cong qK_2$ or $K_3$.
- Derived bounds for $r(C_4,G)$ involving the number of vertices and edges.

## Abstract

Harary's conjecture $r(C_3,G)\leq 2q+1$ for every isolated-free graph G with $q$ edges was proved independently by Sidorenko and Goddard and Klietman. In this paper instead of $C_3$ we consider $K_{2,k}$ and seek a sharp upper bound for $r(K_{2,k},G)$ over all graphs $G$ with $q$ edges. More specifically if $q\geq 2$, we will show that $r(C_4,G)\leq kq+1$ and that equality holds if $G \cong qK_2$ or $K_3$. Using this we will generalize this result for $r(K_{2,k},G)$ when $k>2$. We will also show that for every graph $G$ with $q \geq 2$ edges and with no isolated vertices, $r(C_4, G) \leq 2p+ q - 2$ where $p=|V(G)|$ and that equality holds if $G \cong K_3$.