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

The venue for seminars this term is Room 346 at the SSEES Bulding, 14–16 Taviton Street.

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Next Speaker

• 26th January 2026 - Joshua Erde (University of Birmingham)

  Long paths in the percolated hypercube

  
  

  Let $Q^n$ be the $n$-dimensional binary hypercube. We consider a random subgraph $Q^n_p$ of the hypercube formed by retaining each edge of $Q^n$ independently with probability $p$. Like the binomial random graph $G(n,p)$, this model undergoes a striking phase-transition when $p$ increases beyond the critical value of $1/n$. However, whilst the structure of $G(n,p)$ in this supercritical regime is well understood, much less is known about the structure of $Q^n_p$.

  Confirming a long-standing folklore conjecture, stated in particular by Condon, Espuny Díaz, Girão, Kühn, and Osthus, we show that for any constant $\epsilon >0$ there is a $C:=C(\epsilon)$ such that if $p= C/n$, then with high probability $Q^n_p$ contains a path covering all but an $\epsilon$-fraction of its vertices.

  This is joint work with Sahar Diskin, Joshua Erde, Mihyun Kang, Michael Krivelevich and Lyuben Lichev.