Stability for Intersecting Families in PGL(2,q)

Abstract

We consider the action of the $2$-dimensional projective general linear group $PGL(2,q)$ on the projective line $PG(1,q)$. A subset $S$ of $PGL(2,q)$ is said to be an intersecting family if for every $g_1,g_2 \in S$, there exists $\alpha \in PG(1,q)$ such that $\alpha^{g_1}= \alpha^{g_2}$. It was proved by Meagher and Spiga that the intersecting families of maximum size in $PGL(2,q)$ are precisely the cosets of point stabilizers. We prove that if an intersecting family $S \subset PGL(2,q)$ has size close to the maximum then it must be "close" in structure to a coset of a point stabilizer. This phenomenon is known as stability. We use this stability result proved here to show that if the size of $S$ is close enough to the maximum then $S$ must be contained in a coset of a point stabilizer.

Characterization of intersecting families of maximum size in PSL(2,q)

“…In fact, the size of extremal families in P GL(2, q) is q(q − 1) and every extremal family is a coset of a point stabilizer. Recently, in [24] it was proved that the extremal families in P GL(2, q) are stable.…”

“…The stability of intersecting families has been studied during the past few years (cf. [7,8,24]). Consider the action of S n on [n].…”

Boolean Function Analysis on High-Dimensional Expanders

“…Traditionally, it studies functions on the Boolean cube {−1, 1} n . Recently, the scope of Boolean function analysis has been extended further, encompassing groups [22][23][24]52], association schemes [16, 28-31, 42, 49], error-correcting codes [6], and quantum Boolean functions [47]. Boolean function analysis on extended domains has led to progress in learning theory [49] and on the unique games conjecture [7,16,17,41,42].…”