Cross $t$-intersecting families for symplectic polar spaces

TL;DR

This paper investigates the structure of cross $t$-intersecting families within symplectic polar spaces, establishing that maximum product families are trivial and characterizing non-trivial maximum families.

Contribution

It proves that cross $t$-intersecting families with maximum size product are trivial and describes the structure of non-trivial maximum families in symplectic polar spaces.

Findings

Maximum product cross $t$-intersecting families are trivial.

Characterization of non-trivial maximum families.

Structural insights into symplectic polar space families.

Abstract

Let $\mathscr{P}$ be a symplectic polar space over a finite field $\mathbb{F}_q$, and $\mathscr{P}_m$ denote the collection of all $k$-dimensional totally isotropic subspace in $\mathscr{P}$. Let $\mathscr{F}_1\subset\mathscr{P}_{m_1}$ and $\mathscr{F}_2\subset\mathscr{P}_{m_2}$ satisfy $\dim(F_1\cap F_2)\ge t$ for any $F_1\in\mathscr{F}_1$ and $F_2\in\mathscr{F}_2$. We say they are cross $t$-intersecting families. Moreover, we say they are trivial if each member of them contains a fixed $t$-dimensional totally isotropic subspace. In this paper, we show that cross $t$-intersecting families with maximum product of sizes are trivial. We also describe the structure of non-trivial $t$-intersecting families with maximum product of sizes.