Bollobás-type inequalities for subspaces via weight invariance

Abstract

Let $V$ be an $n$-dimension real vector space with a direct sum decomposition $V=V_1\oplus \cdots \oplus V_r$. Let $\mathcal{P}=\{(A_i,B_i):i\in [m]\}$ be a skew Bollobás system of subspaces of $V$ such that each $i\in [m]$, $A_i={⨁}_{k=1}^r(A_i\cap V_k)$ and $B_i={⨁}_{k=1}^r(B_i\cap V_k)$. We prove that $$\sum _{i=1}^m\prod _{k=1}^r\left[\left(\genfrac{}{}{0ex}{}{a_{i,k}+b_{i,k}}{a_{i,k}}\right)(1+a_{i,k}+b_{i,k}{)}^{-1}\right]\le 1,$$ where $a_{i,k}=\dim (A_i\cap V_k)$ and $b_{i,k}=\dim (B_i\cap V_k)$. This extends a recent result of Yue from set systems to finite dimensional subspaces. We then consider Tuza's theorem on weak Bollobás system for $d$-tuples. We give an alternative proof of the original set version of Tuza, and also establish its vector space analogue. Precisely, let $\mathcal{P}=\{(A_i^{(1)},\dots ,A_i^{(d)}):i\in [m]\}$ be a skew Bollobás system of $d$-tuples of subspaces of finite dimensional space $V$ with $a_i^{(\ell )}=\dim (A_i^{(\ell )})$. Then, for any positive real numbers $p_1,\dots ,p_d$ satisfying $p_1+\cdots +p_d=1$, we prove that ${\sum }_{i=1}^m{\prod }_{\ell =1}^d{p}_{\ell }^{a_i^{(\ell )}}\le 1.$

11 pages