Ramsey properties of maximal (outer)planar graphs

Abstract

We study a natural extension of Ramsey theory relative to the classes of maximally planar and maximally outerplanar graphs. This can be seen as a continuation of the study of `Planar Ramsey theory', introduced by Axenovich et al. The question we ask is the following: For a fixed family $\mathcal{K}$ of graphs and a pair of graphs $\{H,F\}$, does there exist an integer $r_{\mathcal{K}} (H, F)$ such that for every graph $G \in \mathcal{K}$ with $|G| \geq r_{\mathcal{K}}(H, F)$, every red/blue edge-colouring of $G$ admits a red copy of $H$ or a blue copy of $F$? When such an integer exists, we say $\{H,F\}$ is unavoidable in $\mathcal{K}$,, and otherwise $\{H,F\}$ is avoidable in $\mathcal{K}$. Our work focuses on this problem where $\mathcal{K} = \mathcal{K}_{\mathrm{MOP}}$ and $\mathcal{K} = \mathcal{K}_{\mathrm{MP}}$, which denote the families of maximal outerplanar (MOP) graphs and maximal planar (MP) graphs, respectively. This framework generalises the classical Ramsey problem relative to these classes, as the case with $\mathcal{K} = \{K_n \colon n \geq 2\}$ corresponds to classical Ramsey. We also study the corresponding Ramsey numbers for MOP and MP, which we denote as $r_{\mathrm{MOP}}(H, F)$ and $r_{\mathrm{MP}}(H, F)$. In the case when $\mathcal{K} = \mathcal{K}_{\mathrm{MOP}}$, we completely determine all unavoidable pairs $\{H, F\}$ with $|E(F)| \geq 2$, together with upper bounds and sometimes exact values of $r_{\mathrm{MOP}}(H, F)$. When $\mathcal{K} = \mathcal{K}_{\mathrm{MP}}$, we completely determine all unavoidable pairs in the diagonal case $\{H, H\}$ when $H$ is connected, showing that $H$ must be one of the graphs $P_3$, $P_4$, $P_5$, $K_{1, 3}$ or the fork graph $S_{2,1,1}$. This work opens up further possibilities in the study of Ramsey theory relative to a class, and we offer several open problems in this vein.