# Erd\H{o}s-Ko-Rado theorems on the weak Bruhat lattice}

## TL;DR

This paper extends Erd ext{"o}s-Ko-Rado theorems to the weak Bruhat lattice, characterizing maximum intersecting families of permutations and their levels, generalizing classical combinatorial intersection results.

## Contribution

It provides the first characterization of maximum intersecting families of permutations in the weak Bruhat lattice, including level-specific results for large n.

## Key findings

- Characterization of maximum intersecting families of permutations in the Bruhat lattice
- Extension of Erd ext{"o}s-Ko-Rado theorems to permutation lattices
- Results for maximum intersecting families within specific levels for large n

## Abstract

Let ${\mathscr L}=(X,\preceq)$ be a lattice. For ${\cal P}\subseteq X$ we say that ${\cal P}$ is $t$-{\it intersecting} if ${\sf rank}(x\wedge y)\ge t$ for all $x,y\in{\cal P}$. The seminal theorem of Erd\H{o}s, Ko and Rado describes the maximum intersecting ${\cal P}$ in the lattice of subsets of a finite set with the additional condition that ${\cal P}$ is contained within a level of the lattice. The Erd\H{o}s-Ko-Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the $r^{\textrm{th}}$ level of the Bruhat lattice of permutations of size $n$, provided that $n$ is large relative to $r$.