Bollob\'{a}s-type inequalities for subspaces via weight invariance

TL;DR

This paper extends Bollobás-type inequalities to subspaces in finite-dimensional vector spaces, providing new bounds and alternative proofs for vector space analogues of classical combinatorial theorems.

Contribution

It generalizes recent set system inequalities to vector spaces and establishes vector space versions of Tuza's theorem for $d$-tuples of subspaces.

Findings

Derived a new inequality for skew Bollobás systems of subspaces.

Provided an alternative proof of Tuza's theorem in the vector space setting.

Established bounds for $d$-tuples of subspaces with weighted sums.

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 Bollob\'as system of subspaces of $V$ such that each $i\in [m]$, $ A_i = \bigoplus_{k=1}^r (A_i \cap V_k)$ and $ B_i = \bigoplus_{k=1}^r (B_i \cap V_k)$. We prove that $$\sum_{i=1}^{m} \prod_{k=1}^{r} \left[ \binom{a_{i,k} + b_{i,k}}{a_{i,k}} (1 + a_{i,k} + b_{i,k})^{-1} \right] \leq 1,$$ 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 Bollob\'as 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 $\mathcal{P} = \{(A_i^{(1)}, \ldots, A_i^{(d)}) : i \in [m]\}$ be a skew Bollob\'as system of $d$-tuples of subspaces of finite dimensional space $V$ with $a^{(\ell)}_i=\dim (A_i^{(\ell)})$. Then, for any positive real numbers $p_1, \ldots, p_d$ satisfying $p_1 + \cdots + p_d = 1$, we prove that $ \sum_{i=1}^{m} \prod_{\ell=1}^{d} p_{\ell}^{a_i^{(\ell)}} \leq 1. $