Subspace variations of the weighted skew Bollobás theorem

Abstract

Let $V$ be a finite-dimensional real vector space. A collection $\mathcal{P}=\{(A_i,B_i){\}}_{i=1}^m$ of pairs of subspaces of $V$ is called a skew Bollobás system if $\dim (A_i\cap B_i)=0$ for each $i\in [m]$ and $\dim (A_i\cap B_j)>0$ for all $1\le i<j\le m$. Assume that $V=V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}=\{(A_i,B_i){\}}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ satisfying $A_i={⨁}_{k=1}^r(A_i\cap V^{(k)})$ and $B_i={⨁}_{k=1}^r(B_i\cap V^{(k)})$ for each $i\in [m]$. Denote $a_{i,k}=\dim (A_i\cap V^{(k)})$ and $b_{i,k}=\dim (B_i\cap V^{(k)})$. Suppose that $a_{1,k}\le \cdots \le a_{m,k}$ and $b_{1,k}\ge \cdots \ge b_{m,k}$ for each $k\in [r]$. Using the exterior algebraic method developed by Lovász and Scott--Wilmer, we prove that $$\sum _{i=1}^m\frac{1}{{\prod }_{k=1}^r\left(\genfrac{}{}{0ex}{}{a_{i,k}+b_{i,k}}{a_{i,k}}\right)}\le 1.$$ This generalizes the results of Alon (JCTA, 1985) and Scott--Wilmer (JLMS, 2021) to multipart weighted setting. Secondly, we solve a conjecture of Hegedüs (AJC, 2015) concerning projective subspaces, showing that any skew Bollobás system of projective subspaces in an $n$-dimensional projective space contains at most $2^{n+1}-2$ pairs. Thirdly, we prove that if $\mathcal{P}=\{(A_i,B_i){\}}_{i=1}^m$ is a skew Bollobás system of subspaces of $V$ with $a_i=\dim (A_i)$ and $b_i=\dim (B_i)$, then $$\sum _{i=1}^m\frac{1}{(a_i+b_i+1)\left(\genfrac{}{}{0ex}{}{a_i+b_i}{a_i}\right)}\le 1.$$ This gives an extension to the subspace setting of the results of Hegedüs--Frankl (EUJC, 2024) and Yue (DM, 2026). Finally, we extend the above inequality to systems of $d$-tuples of subspaces, giving a unified bound that implies the corresponding results for $d$-tuples of subsets.

20 pages