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arXiv · 2605.12030

A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets

Philipp Heering·Klaus Metsch·Vladislav Taranchuk·Zsuzsa Weiner
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Abstract

Let $D=(\mathcal{P},\mathcal{B})$ be a symmetric $(v,k,\lambda)$-design and let $(X,Y)$ be an equinumerous incidence-free pair, with $X\subseteq \mathcal{P}$ and $Y\subseteq \mathcal{B}$. In this note, we give an elementary proof which shows the existence of a perfect matching between $\mathcal{P} \setminus X$ and $\mathcal{B}\setminus Y$ in the incidence graph of $D$. This recovers a result of Spiro, Adriaensen and Mattheus, who already showed this using different arguments for $k\geq 36$. We use this to connect some dots in the literature and prove that finding the chromatic number of the Kneser graph on chambers of a projective plane is equivalent to finding the incidence-free number of the incidence graph of the plane. Furthermore, we construct an incidence-free pair for PG$(2,q^2)$ of size roughly $q^3/2+3q^2/4$.

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Philipp Heering, Klaus Metsch, Vladislav Taranchuk, Zsuzsa Weiner. 2026-05-12. A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets. https://arxiv.org/abs/2605.12030

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