A variant of the Erdős-Gyárfás problem for $K_8$

Fredy Yip

A variant of the Erdős-Gyárfás problem for $K_8$

Fredy Yip

TL;DR

The paper tackles the Erdős-Gyárfás-inspired problem of colouring edges of $K_n$ with $n^{o(1)}$ colours to avoid even-chromatic copies of a fixed clique $H$, focusing on the challenging case $H=K_8$. It develops an inductive amalgamation framework, using structures like $c\otimes d$ and a rectangle-lemma to control potential obstructions, and proves subpolynomial colour constructions for $K_4$-unique, $K_5$-unique, and $K_8$-odd variants. The main results include a construction of an $n^{o(1)}$-colour edge-colouring avoiding even-chromatic copies of $K_8$, along with stronger unique-chromatic variants for $K_4$ and $K_5$ that also improve known colour bounds for the original problems. These methods advance understanding of linear graph codes and Ramsey-type colourings, and provide a tractable route toward subpolynomial bounds for clique-avoidance colourings in dense graphs.

Abstract

Recently, Alon initiated the study of graph codes and their linear variants in analogy to the study of error correcting codes in theoretical computer science. Alon related the maximum density of a linear graph code which avoids images of a small graph $H$ to the following variant of the Erdős-Gyárfás problem on edge-colourings of $K_n$. A copy of $H$ in an edge-colouring of $K_n$ is even-chromatic if each colour occupies an even number of edges in the copy. We seek an edge-colouring of $K_n$ using $n^{o(1)}$ colours such that there are no even-chromatic copies of $H$. Such an edge-colouring is conjectured to exist for all cliques $K_t$ with an even number of edges. To date, edge-colourings satisfying this property have been constructed for $K_4$ and $K_5$. We construct an edge-colouring using $n^{o(1)}$ colours which avoids even-chromatic copies of $K_8$. This was the smallest open case of the above conjecture, as $K_6, K_7$ each has an odd number of edges. We also study a stronger condition on edge-colourings, where for each copy of $H$, there is a colour occupying exactly one edge in the copy. We conjecture that an edge-colouring using $n^{o(1)}$ colours and satisfying this stronger requirement exists for all cliques $K_t$ regardless of the parity of the number of its edges. We construct edge-colourings satisfying this stronger property for $K_4$ and $K_5$. These constructions also improve upon the number of colours needed for the original problem of avoiding even-chromatic copies of $K_4$ and $K_5$.

A variant of the Erdős-Gyárfás problem for $K_8$

TL;DR

The paper tackles the Erdős-Gyárfás-inspired problem of colouring edges of

with

colours to avoid even-chromatic copies of a fixed clique

, focusing on the challenging case

. It develops an inductive amalgamation framework, using structures like

and a rectangle-lemma to control potential obstructions, and proves subpolynomial colour constructions for

-unique,

-unique, and

-odd variants. The main results include a construction of an

-colour edge-colouring avoiding even-chromatic copies of

, along with stronger unique-chromatic variants for

and

that also improve known colour bounds for the original problems. These methods advance understanding of linear graph codes and Ramsey-type colourings, and provide a tractable route toward subpolynomial bounds for clique-avoidance colourings in dense graphs.

Abstract

to the following variant of the Erdős-Gyárfás problem on edge-colourings of

. A copy of

in an edge-colouring of

is even-chromatic if each colour occupies an even number of edges in the copy. We seek an edge-colouring of

using

colours such that there are no even-chromatic copies of

. Such an edge-colouring is conjectured to exist for all cliques

with an even number of edges. To date, edge-colourings satisfying this property have been constructed for

and

. We construct an edge-colouring using

colours which avoids even-chromatic copies of

. This was the smallest open case of the above conjecture, as

each has an odd number of edges. We also study a stronger condition on edge-colourings, where for each copy of

, there is a colour occupying exactly one edge in the copy. We conjecture that an edge-colouring using

colours and satisfying this stronger requirement exists for all cliques

regardless of the parity of the number of its edges. We construct edge-colourings satisfying this stronger property for

and

. These constructions also improve upon the number of colours needed for the original problem of avoiding even-chromatic copies of

and

Paper Structure (8 sections, 17 theorems, 36 equations)

This paper contains 8 sections, 17 theorems, 36 equations.

Introduction
Inductive colouring construction
General Observations
A $K_4$-unique edge-colouring with $n^{o(1)}$ colours
A $K_5$-unique edge-colouring with $n^{o(1)}$ colours
A $K_8$-odd edge-colouring with $n^{o(1)}$ colours
Bounds on colour usage
A proof of Proposition \ref{['non-clique implies non-unique']}

Key Result

Theorem 1.7

If $H$ is even-decomposable, then $r_H(n) = n^{\Omega(1)}$.

Theorems & Definitions (30)

Theorem 1.7: Versteegen versteegen2023upper
Conjecture 1.8: Versteegen versteegen2023upper
Conjecture 1.9: Ge, Xu and Zhang ge2023new
Theorem 1.10: Cameron and Heath cameron2023new
Theorem 1.11: Bennett, Heath and Zerbib heath2023edge; Ge, Xu and Zhang ge2023new
Theorem 1.12
Corollary 1.13
Proposition 1.15
Proposition 1.16
Proposition 1.18
...and 20 more

A variant of the Erdős-Gyárfás problem for $K_8$

TL;DR

Abstract

A variant of the Erdős-Gyárfás problem for $K_8$

Authors

TL;DR

Abstract

Table of Contents

Key Result

Theorems & Definitions (30)