$s$-almost cross-$t$-intersecting families for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field $\mathbb{F} _{q} $, and ${V\brack k}$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F},\mathcal{G}\subseteq {V\brack k}$ are said to be cross-$t$-intersecting if $\dim(F\cap G)\ge t$ for all $F\in \mathcal{F}, G\in \mathcal{G}$. Two families $\mathcal{F}$ and $\mathcal{G}$ are called $s$-almost cross-$t$-intersecting if each member of $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members of $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we discribe the structure of $s$-almost cross-$t$-intersecting families with maximum product of their sizes. In addition, we prove a stability result.

💡 Research Summary

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The paper investigates families of k‑dimensional subspaces of an n‑dimensional vector space V over a finite field (\mathbb{F}_q). Two families (\mathcal{F},\mathcal{G}\subseteq{V\brack k}) are called cross‑t‑intersecting if every pair (F\in\mathcal{F}, G\in\mathcal{G}) satisfies (\dim(F\cap G)\ge t). The authors weaken this condition by allowing each subspace to be t‑disjoint with at most s members of the opposite family; such families are termed s‑almost cross‑t‑intersecting.

The main contributions are two structural theorems. Theorem 1.1 establishes that, provided (k\ge t+1) and (n\ge 2k+2t+1+\log_q7s), any s‑almost cross‑t‑intersecting pair satisfies
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