The University College London (UCL) Combinatorics Seminar is held every Monday at 4–5pm during term time.

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Next Speaker

• 2nd June 2025 - Benedict Randall Shaw (University of Cambridge)

  Hypergraph jumps and non-jumps

  
  

  Venue: LG16 Seminar Room, Bentham House.

  An $r$-uniform hypergraph, or $r$-graph, has density $|E(G)|/\big|V(G)^{(r)}\big|$. We say $\alpha$ is a jump for $r$-graphs if there is some constant $\delta=\delta(\alpha)$ such that, for each $\varepsilon>0$ and $n\geqslant r$, any sufficiently large $r$-graph of density at least $\varepsilon$ has a subgraph of order $n$ and density at least $\alpha+\delta$. For $r=2$, all $\alpha$ are jumps. For $r\geqslant 3$, Erdős showed all $\big[0,\frac{r!}{r^r}\big)$ are jumps, and conjectured all $[0,1)$ are jumps. Since then, a variety of non-jumps have been proven, using a method introduced by Frankl and Rödl.

  Our aim is to provide a general setting for this method. As an application, we prove several new non-jumps, which are smaller than any previously known. We also demonstrate that these are the smallest the current method can prove.