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arXiv · 2402.17687

On the maximum intersecting sets of the general semilinear group of degree $2$

Roghayeh Maleki·Andriaherimanana Sarobidy Razafimahatratra
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Abstract

Let $p$ be a prime and $q = p^k$. A subset $\mathcal{F} \subset \operatorname{\Gamma L}_{2}(q)$ is intersecting if any two semilinear transformations in $\mathcal{F}$ agree on some non-zero vector in $\mathbb{F}_q^2$. We show that any intersecting set of $\operatorname{\Gamma L}_{2}(q)$ is of size at most that of a stabilizer of a non-zero vector, and we characterize the intersecting sets of this size. Our proof relies on finding a subgraph which is a lexicographic product in the derangement graph of $\operatorname{\Gamma L}_{2}(q)$ in its action on non-zero vectors of $\mathbb{F}_q^2$. This method is also applied to give a new proof that the only maximal intersecting sets of $\operatorname{GL}_{2}(q)$ are the maximum intersecting sets.

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Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra. 2024-02-27. On the maximum intersecting sets of the general semilinear group of degree $2$. https://arxiv.org/abs/2402.17687

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