Relative discrepancy of hypergraphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912456332279808 |
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| author | Luong-Le, Diep Tran, Tuan Yang, Dilong |
| author_facet | Luong-Le, Diep Tran, Tuan Yang, Dilong |
| contents | Given $k$-uniform hypergraphs $G$ and $H$ on $n$ vertices with densities $p$ and $q$, their relative discrepancy is defined as $\hbox{disc}(G,H)=\max\big||E(G')\cap E(H')|-pq\binom{n}{k}\big|$, where the maximum ranges over all pairs $G',H'$ with $G'\cong G$, $H'\cong H$, and $V(G')=V(H')$. Let $\hbox{bs}(k)$ denote the smallest integer $m \ge 2$ such that any collection of $m$ $k$-uniform hypergraphs on $n$ vertices with moderate densities contains a pair $G,H$ for which $\hbox{disc}(G,H) = Ω(n^{(k+1)/2})$.
In this paper, we answer several questions raised by Bollobás and Scott, providing both upper and lower bounds for $\hbox{bs}(k)$. Consequently, we determine the exact value of $\hbox{bs}(k)$ for $2\le k\le 13$, and show $\hbox{bs}(k)=O(k^{0.525})$, substantially improving the previous bound $\hbox{bs}(k)\le k+1$ due to Bollobás-Scott. The case $k=2$ recovers a result of Bollobás-Scott, which generalises classical theorems of Erdős-Spencer, and Erdős-Goldberg-Pach-Spencer. The case $k=3$ also follows from the results of Bollobás-Scott and Kwan-Sudakov-Tran. Our proof combines linear algebra, Fourier analysis, and extremal hypergraph theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_23264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Relative discrepancy of hypergraphs Luong-Le, Diep Tran, Tuan Yang, Dilong Combinatorics Given $k$-uniform hypergraphs $G$ and $H$ on $n$ vertices with densities $p$ and $q$, their relative discrepancy is defined as $\hbox{disc}(G,H)=\max\big||E(G')\cap E(H')|-pq\binom{n}{k}\big|$, where the maximum ranges over all pairs $G',H'$ with $G'\cong G$, $H'\cong H$, and $V(G')=V(H')$. Let $\hbox{bs}(k)$ denote the smallest integer $m \ge 2$ such that any collection of $m$ $k$-uniform hypergraphs on $n$ vertices with moderate densities contains a pair $G,H$ for which $\hbox{disc}(G,H) = Ω(n^{(k+1)/2})$. In this paper, we answer several questions raised by Bollobás and Scott, providing both upper and lower bounds for $\hbox{bs}(k)$. Consequently, we determine the exact value of $\hbox{bs}(k)$ for $2\le k\le 13$, and show $\hbox{bs}(k)=O(k^{0.525})$, substantially improving the previous bound $\hbox{bs}(k)\le k+1$ due to Bollobás-Scott. The case $k=2$ recovers a result of Bollobás-Scott, which generalises classical theorems of Erdős-Spencer, and Erdős-Goldberg-Pach-Spencer. The case $k=3$ also follows from the results of Bollobás-Scott and Kwan-Sudakov-Tran. Our proof combines linear algebra, Fourier analysis, and extremal hypergraph theory. |
| title | Relative discrepancy of hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2506.23264 |