Subspace variations of the weighted skew Bollobás theorem
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911481787842560 |
|---|---|
| author | Wu, Yongjiang Li, Yongtao Lu, Lu Feng, Lihua |
| author_facet | Wu, Yongjiang Li, Yongtao Lu, Lu Feng, Lihua |
| contents | 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 $1\leq i<j \leq m$. Assume that $V = V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás 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 Lovász 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 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 $$ \sum_{i=1}^m \frac{1}{(a_i+ b_i+1)\binom{a_i+b_i}{a_i}} \le 1. $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_02698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subspace variations of the weighted skew Bollobás theorem Wu, Yongjiang Li, Yongtao Lu, Lu Feng, Lihua Combinatorics 05D05 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 $1\leq i<j \leq m$. Assume that $V = V^{(1)}\oplus \cdots \oplus V^{(r)}$ and $\mathcal{P}= \{(A_i,B_i)\}_{i=1}^m$ is a skew Bollobás 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 Lovász 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 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 $$ \sum_{i=1}^m \frac{1}{(a_i+ b_i+1)\binom{a_i+b_i}{a_i}} \le 1. $$ 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. |
| title | Subspace variations of the weighted skew Bollobás theorem |
| topic | Combinatorics 05D05 |
| url | https://arxiv.org/abs/2603.02698 |