An exponential upper bound for induced Ramsey numbers

TL;DR

This work proves an exponential upper bound for induced multicolor Ramsey numbers by embedding arbitrary $k$-vertex graphs into a random host graph and carefully controlling the appearance of induced monochromatic copies. The authors blend a vertex-by-vertex random embedding strategy with a novel non-Janson container theory and a detailed extension-collection framework to guarantee, with high probability, that every $r$-coloring yields an induced monochromatic copy of any $k$-vertex graph $H$. The central technical innovations include encoding induced copies via hypergraphs, a robust $(p,R)$-Janson formalism, and a general container theorem for non-Janson sets, enabling global success probabilities to be inferred from local obstructions. Consequently, they establish $R_{ind}(H; r) \le r^{C r k}$ for some $C>0$, resolving Erdős's conjecture for $r=2$ and strengthening the multicolor results for all fixed $r$. In addition, the paper shows that almost every $G$ on $N=r^{C r k}$ vertices satisfies the induced Ramsey property for all $k$-vertex graphs, highlighting the practical reach of the approach.

Abstract

The induced Ramsey number $R_{\mathrm{ind}}(H; r)$ of a graph $H$ is the minimum number $N$ such that there exists a graph with $N$ vertices for which all $r$-colourings of its edges contain a monochromatic induced copy of $H$. Our main result is the existence of a constant $C > 0$ such that, for every graph $H$ on $k$ vertices, these numbers satisfy \begin{equation*} R_{\mathrm{ind}}(H; r) \le r^{C r k}. \end{equation*} When $r = 2$, this resolves a conjecture of Erdős from 1975. For $r > 2$, it answers a question of Conlon, Fox and Sudakov in a strong form.

Theorems & Definitions (91)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• proof
• proof
• Definition 2.5
• Definition 2.5
• Lemma 2.5
• proof : Proof that \ref{['stmt:key']} implies \ref{['stmt:multicolour']}