Induced subgraph density. VII. The five-vertex path

TL;DR

This work proves the Erdős-Hajnal property for the five-vertex path $P_5$ by establishing a polynomial Rödl form for $P_5$-free graphs. The authors blend probabilistic and structural methods within an iterative sparsification framework, using blockade-based decompositions and comb constructions to build long, pure or sparsely connected blockades. A two-part argument first produces a robust blockade and then extracts stronger structure to derive the main polynomial Rödl bound, culminating in the $P_5$ Erdős-Hajnal result. The results advance the understanding of induced-subgraph structure in $P_5$-free graphs and reinforce the connection between the Erdős-Hajnal and Fox–Sudakov conjectures through a constructive blockade methodology.

Abstract

We prove the Erdős-Hajnal conjecture for the five-vertex path $P_5$; that is, there exists $c>0$ such that every $n$-vertex graph with no induced $P_5$ has a clique or stable set of size at least $n^c$. This completes the verification of the Erdős-Hajnal property of all five-vertex graphs. Our methods combine probabilistic and structural ideas with the iterative sparsification framework introduced in the third and fourth papers in the series.

Figures (2)

• Figure 1: Making a house from an upside-down comb with anticonnected blocks.
• Figure 2: Using a really long semisparse blockade.

Theorems & Definitions (32)

• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Theorem 1.5
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Lemma 3.4
• Lemma 4.1