Rainbow triangles and the Erdős-Hajnal problem in projective geometries
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| Format: | Preprint |
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2025
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| _version_ | 1866917017587548160 |
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| author | Chun, Carolyn Douthitt, James Dylan Ge, Wayne Huynh, Tony Kroeker, Matthew E. Nelson, Peter |
| author_facet | Chun, Carolyn Douthitt, James Dylan Ge, Wayne Huynh, Tony Kroeker, Matthew E. Nelson, Peter |
| contents | 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 $(k,q) = (3,2)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13781 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rainbow triangles and the Erdős-Hajnal problem in projective geometries Chun, Carolyn Douthitt, James Dylan Ge, Wayne Huynh, Tony Kroeker, Matthew E. Nelson, Peter Combinatorics 05B35, 05C15 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 $(k,q) = (3,2)$. |
| title | Rainbow triangles and the Erdős-Hajnal problem in projective geometries |
| topic | Combinatorics 05B35, 05C15 |
| url | https://arxiv.org/abs/2505.13781 |