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On polynomially high-chromatic pure pairs

Tung H. Nguyen

TL;DR

This work advances the polynomial Gyárfás–Sumner program by developing structural tools for $T$-free graphs, emphasizing $P_5$-free graphs. It combines expository and technical results: a path-augmenting approach via covering blockades, a framework of controlled induced subgraphs and terminal partitions, and an addibility result showing that disjoint unions of $t$-brooms preserve polynomial $\chi$-bounding. A central achievement is a polynomial-vs-linear complete-pair bound for $P_5$-free graphs, yielding $\chi(G) \le w^{O(\log w/\log\log w)}$ when $\omega(G)=w$, and a general near-pure-pair theorem for arbitrary forbidden graphs $H$, with broader implications for the Erdős–Hajnal phenomenon and for poly-$\chi$-bounding in forest-avoidance classes.

Abstract

Let $T$ be a forest. We study polynomially high-chromatic pure pairs in graphs with no $T$ as an induced subgraph ($T$-free graphs in other words), with applications to the polynomial Gyárfás-Sumner conjecture. In addition to reproving several known results in the literature, we deduce: $\bullet$ If $T=P_5$ is the five-vertex path, then every $T$-free graph $G$ with clique number $w\ge2$ contains a complete pair $(A,B)$ of induced subgraphs with $χ(A)\ge w^{-d}χ(G)$ and $χ(B)\ge 2^{-d}χ(G)$, for some universal $d\ge1$. The proof uses the recent Erdős-Hajnal result for $P_5$-free graphs. Via the classical Gyárfás path argument, such a ``polynomial versus linear high-$χ$ complete pairs'' result can be viewed as further supporting evidence for the polynomial Gyárfás-Sumner conjecture for $P_5$. In particular, it implies \[χ(G)\le w^{O(\log w/\log\log w)}\] which asymptotically improves the bound $χ(G)\le w^{\log w}$ of Scott, Seymour, and Spirkl. $\bullet$ If $T$ and a broom satisfy the polynomial Gyárfás-Sumner conjecture, then so does their disjoint union. Unifying earlier results of Chudnovsky, Scott, Seymour, and Spirkl, and of Scott, Seymour, and Spirkl, this gives new instances of $T$ for which the conjecture holds.

On polynomially high-chromatic pure pairs

TL;DR

This work advances the polynomial Gyárfás–Sumner program by developing structural tools for -free graphs, emphasizing -free graphs. It combines expository and technical results: a path-augmenting approach via covering blockades, a framework of controlled induced subgraphs and terminal partitions, and an addibility result showing that disjoint unions of -brooms preserve polynomial -bounding. A central achievement is a polynomial-vs-linear complete-pair bound for -free graphs, yielding when , and a general near-pure-pair theorem for arbitrary forbidden graphs , with broader implications for the Erdős–Hajnal phenomenon and for poly--bounding in forest-avoidance classes.

Abstract

Let be a forest. We study polynomially high-chromatic pure pairs in graphs with no as an induced subgraph (-free graphs in other words), with applications to the polynomial Gyárfás-Sumner conjecture. In addition to reproving several known results in the literature, we deduce: If is the five-vertex path, then every -free graph with clique number contains a complete pair of induced subgraphs with and , for some universal . The proof uses the recent Erdős-Hajnal result for -free graphs. Via the classical Gyárfás path argument, such a ``polynomial versus linear high- complete pairs'' result can be viewed as further supporting evidence for the polynomial Gyárfás-Sumner conjecture for . In particular, it implies which asymptotically improves the bound of Scott, Seymour, and Spirkl. If and a broom satisfy the polynomial Gyárfás-Sumner conjecture, then so does their disjoint union. Unifying earlier results of Chudnovsky, Scott, Seymour, and Spirkl, and of Scott, Seymour, and Spirkl, this gives new instances of for which the conjecture holds.
Paper Structure (13 sections, 29 theorems, 37 equations)

This paper contains 13 sections, 29 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Some expository arguments
  3. Excluding a $t$-broom
  4. A general result
  5. New confirmed cases of polynomial Gyárfás--Sumner
  6. Adding a path via covering blockades
  7. Controlled induced subgraphs
  8. Adding a broom
  9. Polynomial versus linear complete pairs in $P_5$-free graphs
  10. Basic facts
  11. Colourful induced subgraphs
  12. Terminal partitions in controlled $P_5$-free graphs
  13. Colourful induced subgraphs versus high-$\chi$ pure pairs

Key Result

Theorem 1.3

There exists $a\ge4$ such that for every $k\ge1$, every $P_5$-free graph with more than $k^a$ vertices has a clique or stable set with more than $k$ vertices.

Theorems & Definitions (41)

  • Conjecture 1.1: Gyárfás--Sumner
  • Conjecture 1.2: Polynomial Gyárfás--Sumner
  • Theorem 1.3: Nguyen--Scott--Seymour
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6: Gyárfás--Sumner
  • Theorem 1.7: Scott--Seymour--Spirkl
  • Theorem 1.8: Chudnovsky--Scott--Seymour--Spirkl
  • Theorem 1.9
  • Theorem 1.10
  • ...and 31 more