Non-trivial $t$-intersecting families for symplectic polar spaces

TL;DR

This paper characterizes the structure of the largest non-trivial $t$-intersecting families of $m$-dimensional subspaces in symplectic polar spaces over finite fields, extending understanding beyond trivial intersecting families.

Contribution

It determines the structure of maximum-sized non-trivial $t$-intersecting subfamilies in symplectic polar spaces, a significant extension of intersecting family theory.

Findings

Identifies the structure of maximum non-trivial $t$-intersecting families.

Distinguishes between trivial and non-trivial intersecting families.

Provides a classification for maximum non-trivial intersecting families.

Abstract

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the set of all $m$-dimensional subspaces in $\mathscr{P}$. We say a $t$-intersecting subfamily of $\mathscr{P}_m$ is trivial if there exists a $t$-dimensional subspace contained in each member of this family. In this paper, we determine the structure of maximum sized non-trivial $t$-intersecting subfamilies of $\mathscr{P}_m$.