New results of Bollobás-type theorem for affine subspaces and projective subspaces

TL;DR

The paper resolves the exact maximal size of cross-intersecting affine subspace pairs over a finite field: for any prime power $q$ and dimension $n\ge 1$, the maximum is $m(n,q)=2\cdot\frac{q^{n}-1}{q-1}$, demonstrated by a tight hyperplane-based construction and a matching upper bound. It also provides a bound for cross-intersecting projective subspaces and, in the case $q=2$, confirms Hegedüs' conjecture with $m\le 2^{n+1}-2$ via a polynomial-orthogonality argument using projective-point characteristic vectors. Together, these results advance Bollobás-type theorems in affine and projective geometries, answer open questions for $q=2$, and clarify the landscape for $q\neq 2$, while highlighting open problems and directions for future work in the general case.

Abstract

Bollobás-type theorem has received a lot of attention due to its application in graph theory. In 2015, Gábor Heged{ü}s gave an upper bound of bollobás-type affine subspace families for $q\neq 2$, and constructed an almost sharp affine subspaces pair families. In this note, we prove a new upper bound for bollobás-type affine subspaces without the requirement of $q\neq 2$, and construct a pair of families of affine subspaces, which shows that our upper bound is sharp. We also give an upper bound for bollobás-type projective subspaces, and prove that the Heged{ü}s's conjecture holds when $q=2$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Theorem 2.1
• proof
• Theorem 2.2
• proof