Rainbow triangles and the Erdős-Hajnal problem in projective geometries

TL;DR

This work extends the Erdős–Hajnal paradigm to finite projective geometries, formulating multicolour and induced variants in PG$(n-1,q)$ and resolving key binary cases for the triangle. It develops a modular matroid framework with lift-joins and targets to obtain a structural decomposition of rainbow-triangle-free colourings, yielding both nonrainbow and rainbow results, including a Gallai-type structure for rainbow triangles and near-sharp lower bounds via probabilistic constructions. The paper also establishes resolutions for several small $(k,q)$ instances and demonstrates limits by constructing infinite-field counterexamples using non-Archimedean valuations, highlighting the boundaries of current structure theorems. Overall, it provides a coherent toolkit (modularity, lift-joins, targets) to derive large homogeneous subspaces in geometric settings and clarifies where finite-field techniques fundamentally fail in the infinite-field regime.

Abstract

We formulate a geometric version of the Erdős-Hajnal conjecture that applies to finite projective geometries rather than graphs, in both its usual 'induced' form and the multicoloured form. The multicoloured conjecture states, roughly, that a colouring $c$ of the points of $\mathsf{PG}(n-1,q)$ containing no copy of a fixed colouring $c_0$ of $\mathsf{PG}(k-1,q)$ for small $k$ must contain a subspace of dimension polynomial in $n$ that avoids some colour. If $(k,q) = (2,2)$, then $c_0$ is a colouring of a three-element 'triangle', and there are three essentially different cases, all of which we resolve. We derive both the cases where $c_0$ assigns the same colour to two different elements from a recent breakthrough result in additive combinatorics due to Kelley and Meka. We handle the case that $c_0$ is a 'rainbow' colouring by proving that rainbow-triangle-free colourings of projective geometries are exactly those that admit a certain decomposition into two-coloured pieces. This is closely analogous to a theorem of Gallai on rainbow-triangle-free coloured complete graphs. We also show that existing structure theorems resolve certain two-coloured cases where $(k,q) = (2,3)$, and $(k,q) = (3,2)$.

Theorems & Definitions (136)

• Conjecture 1.1: Multicoloured Erdős-Hajnal conjecture for graphs
• Conjecture 1.2: Multicoloured geometric Erdős-Hajnal conjecture
• Conjecture 1.3: Geometric Erdős-Hajnal conjecture
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10