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 Bollob\'as system if $\dim(A_i\cap B_i)=0$ for each $i\in [m]$ and $\dim(A_i\cap B_j)>0$ for all $1\leq i<j \leq m$. Assume that $V = V^{ (1)}\oplus \cdots \oplus V^{ (r)}$ and $\mathcal{ P}= \{ (A_i,B_i)\}_{ i=1}^m$ is a skew Bollob\'as system of subspaces of $V$ satisfying $ A_i = \bigoplus_{ k=1}^r (A_i \cap V^{ (k)})$ and $ B_i = \bigoplus_{ 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 Lov\'{ a}sz and Scott--Wilmer, we prove that $$ \sum_{ i=1}^{ m} \frac{ 1}{ \prod_{ k=1}^{ r} \binom{ a_{ i,k}+b_{ i,k}}{ a_{ i,k}}} \le 1 . $$ This generalizes the results of Alon (JCTA, 1985) and Scott--Wilmer (JLMS, 2021) to multipart weighted setting. Secondly, we solve a conjecture of Heged\"us (AJC, 2015) concerning projective subspaces, showing that any skew Bollob\'as 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 Bollob\'as system of subspaces of $V$ with $a_i=\dim (A_i)$ and $b_i=\dim (B_i)$, then $$ \sum_{ i=1}^m \frac{ 1}{ (a_i+ b_i+1)\binom{ a_i+b_i}{ a_i}} \le 1. $$ This gives an extension to the subspace setting of the results of Heged\"us--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.
Categories: math.CO