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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2603.12610 |
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| _version_ | 1866908883474186240 |
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| author | Du, Longma Hu, Xinyu Liu, Ruilong Wang, Guanghui |
| author_facet | Du, Longma Hu, Xinyu Liu, Ruilong Wang, Guanghui |
| contents | For $2\le k\le t<s$, the Erdős-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London Math. Soc. 2018) conjectured that $f^{(k)}_{k+1,k+2}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for $k\ge 4$, where $\log_{(i)}$ denotes the $i$-fold iterated logarithm. This is equivalent to the statement that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for every $s\ge k+2$.
In this paper, we introduce multi-color patterns into a random construction of a $2$-graph to build a $4$-graph, and for the first time, combine them with multi-layer extremum structures to prove that $f^{(4)}_{5,s}(N)=(\log \log N)^{Θ(1)}$ for every $s\ge 11$. More generally, using a variant of the Erdős-Hajnal stepping-up lemma, we also establish that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for every $s\ge k+7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_12610 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A step towards the Erdős-Rogers problem Du, Longma Hu, Xinyu Liu, Ruilong Wang, Guanghui Combinatorics For $2\le k\le t<s$, the Erdős-Rogers function $f^{(k)}_{t,s}(N)$ denotes the largest $m$ such that every $K^{(k)}_s$-free $k$-graph on $N$ vertices contains a $K^{(k)}_t$-free induced subgraph on $m$ vertices. Mubayi and Suk (J. London Math. Soc. 2018) conjectured that $f^{(k)}_{k+1,k+2}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for $k\ge 4$, where $\log_{(i)}$ denotes the $i$-fold iterated logarithm. This is equivalent to the statement that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for every $s\ge k+2$. In this paper, we introduce multi-color patterns into a random construction of a $2$-graph to build a $4$-graph, and for the first time, combine them with multi-layer extremum structures to prove that $f^{(4)}_{5,s}(N)=(\log \log N)^{Θ(1)}$ for every $s\ge 11$. More generally, using a variant of the Erdős-Hajnal stepping-up lemma, we also establish that $f^{(k)}_{k+1,s}(N)=(\log_{(k-2)}N)^{Θ(1)}$ for every $s\ge k+7$. |
| title | A step towards the Erdős-Rogers problem |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.12610 |