Minimal hypergraph non-jumps
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911000154865664 |
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| author | Shaw, Benedict Randall |
| author_facet | Shaw, Benedict Randall |
| contents | An $r$-uniform hypergraph, or $r$-graph, has density $|E(G)|/|V(G)^{(r)}|$. We say $α$ is a jump for $r$-graphs if there is some constant $δ=δ(α)$ such that, for each $\varepsilon>0$ and $n\geq r$, any sufficiently large $r$-graph of density at least $\varepsilon$ has a subgraph of order $n$ and density at least $α+δ$. For $r=2$, all $α$ are jumps. For $r\geq 3$, Erdős showed all $[0,\frac{r!}{r^r})$ are jumps, and conjectured all $[0,1)$ are jumps. Since then, a variety of non-jumps have been proved, using a method introduced by Frankl and Rödl.
Our aim in this paper is to provide a general setting for this method. As an application, we give several new non-jumps, which are smaller than any previously known. We also demonstrate that these are the smallest the current method can prove. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal hypergraph non-jumps Shaw, Benedict Randall Combinatorics 05C65 An $r$-uniform hypergraph, or $r$-graph, has density $|E(G)|/|V(G)^{(r)}|$. We say $α$ is a jump for $r$-graphs if there is some constant $δ=δ(α)$ such that, for each $\varepsilon>0$ and $n\geq r$, any sufficiently large $r$-graph of density at least $\varepsilon$ has a subgraph of order $n$ and density at least $α+δ$. For $r=2$, all $α$ are jumps. For $r\geq 3$, Erdős showed all $[0,\frac{r!}{r^r})$ are jumps, and conjectured all $[0,1)$ are jumps. Since then, a variety of non-jumps have been proved, using a method introduced by Frankl and Rödl. Our aim in this paper is to provide a general setting for this method. As an application, we give several new non-jumps, which are smaller than any previously known. We also demonstrate that these are the smallest the current method can prove. |
| title | Minimal hypergraph non-jumps |
| topic | Combinatorics 05C65 |
| url | https://arxiv.org/abs/2506.09620 |