About Füredi's conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910479151005696 |
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| author | Hegedüs, Gábor |
| author_facet | Hegedüs, Gábor |
| contents | Let $t$ be a non-negative integer and $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a set-pair family satisfying $|A_i \cap B_i|\leq t$ for $1\leq i \leq m$. $\mbox{$\cal P$}$ is called strong Bollobás $t$-system, if $|A_i\cap B_j|>t$ for all $1\leq i\neq j \leq m$.
Füredi conjectured the following nice generalization of Bollobás' Theorem:
Let $t$ be a non-negative integer. Let $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a strong Bollobás $t$-system. Then $$ \sum_{i=1}^m \frac{1}{|A_i|+|B_i|-2t \choose |A_i|-t}\leq 1. $$ We confirmed the following special case of Füredi's conjecture along with some more results of similar flavor.
Let $t$ be a non-negative integer. Let $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ denote a strong Bollobás $t$-system. Define $a_i:=|A_i|$ and $b_i:=|B_i|$ for each $i$. Assume that there exists a positive integer $N$ such that $a_i+b_i=N$ for each $i$. Then $$ \sum_{i=1}^m \frac{1}{a_i+b_i-2t \choose a_i-t}\leq 1. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_05841 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | About Füredi's conjecture Hegedüs, Gábor Combinatorics 05D05, 15A75, 15A03 Let $t$ be a non-negative integer and $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a set-pair family satisfying $|A_i \cap B_i|\leq t$ for $1\leq i \leq m$. $\mbox{$\cal P$}$ is called strong Bollobás $t$-system, if $|A_i\cap B_j|>t$ for all $1\leq i\neq j \leq m$. Füredi conjectured the following nice generalization of Bollobás' Theorem: Let $t$ be a non-negative integer. Let $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ be a strong Bollobás $t$-system. Then $$ \sum_{i=1}^m \frac{1}{|A_i|+|B_i|-2t \choose |A_i|-t}\leq 1. $$ We confirmed the following special case of Füredi's conjecture along with some more results of similar flavor. Let $t$ be a non-negative integer. Let $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ denote a strong Bollobás $t$-system. Define $a_i:=|A_i|$ and $b_i:=|B_i|$ for each $i$. Assume that there exists a positive integer $N$ such that $a_i+b_i=N$ for each $i$. Then $$ \sum_{i=1}^m \frac{1}{a_i+b_i-2t \choose a_i-t}\leq 1. $$ |
| title | About Füredi's conjecture |
| topic | Combinatorics 05D05, 15A75, 15A03 |
| url | https://arxiv.org/abs/2406.05841 |