Ramsey numbers of quadrilateral versus books

TL;DR

This paper determines the exact Ramsey numbers for quadrilaterals versus books for infinitely many values, improving previous bounds and providing new exact values especially for prime power cases.

Contribution

The paper introduces a new upper bound for the Ramsey number r(C_4, B_n) and determines exact values for infinitely many n, including prime power cases, advancing understanding of these graph Ramsey numbers.

Findings

Established a new upper bound: r(C_4, B_{(m-1)^2+(t-2)}) ≤ m^2 + t.

Determined exact values of r(C_4, B_n) for infinitely many n.

Proved r(C_4, B_{q^2 - q - 2}) = q^2 + q - 1 for all prime powers q ≥ 4.

Abstract

A book $B_n$ is a graph which consists of $n$ triangles sharing a common edge. In this paper, we study Ramsey numbers of quadrilateral versus books. Previous results give the exact value of $r(C_4,B_n)$ for $1\le n\le 14$. We aim to show the exact value of $r(C_4,B_n)$ for infinitely many $n$. To achieve this, we first prove that $r(C_4,B_{(m-1)^2+(t-2)})\le m^2+t$ for $m\ge4$ and $0 \leq t \leq m-1$. This improves upon a result by Faudree, Rousseau and Sheehan (1978) which states that \begin{align*} r(C_4,B_n)\le g(g(n)), \;\;\text{where}\;\;g(n)=n+\lfloor\sqrt{n-1}\rfloor+2. \end{align*} Combining the new upper bound and constructions of $C_4$-free graphs, we are able to determine the exact value of $r(C_4,B_n)$ for infinitely many $n$. As a special case, we show $r(C_4,B_{q^2-q-2}) = q^2+q-1$ for all prime power $q\ge4$.