Upcoming talks

• 27th October 2025 - Natalie Behague (University of Warwick)

  On random regular graphs and the Kim-Vu Sandwich Conjecture

  
  

  The random regular graph $G_d(n)$ is selected uniformly at random from all $d$-regular graphs on $n$ vertices. This model is a lot harder to study than the Erdős-Renyi binomial random graph model $G(n, p)$ as the probabilities of edges being present are not independent. However, in the regime $d \gg \log n$, various graph properties including Hamiltonicity and chromatic number were shown (with hard work) to be the same in $G_d(n)$ as in $G(n, p)$ with $np = d$, which inspired Kim and Vu to conjecture that when $d \gg \log n$ it is possible to ‘sandwich’ the random regular graph $G_d(n)$ between two Erdős-Renyi random graphs with similar edge density. A proof of this sandwich conjecture would unify all the previous separate hard-won results. 

  Various authors have proved weaker versions of the sandwich conjecture with incrementally improved bounds on d — the previous state of the art was due to Gao, Isaev and McKay who proved the conjecture for $d \gg (\log n)^4$. I will sketch our new proof of the full conjecture.

  This is joint work with Richard Montgomery and Daniel Il’kovič.

  
  
  

• 10th November 2025 - Debsoumya Chakraborti (University College London)

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• 17th November 2025 - Noah Kravitz (University of Oxford)

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• 24th November 2025 - Katherine Staden (Open University)

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• 1st December 2025 - Jane Tan (University of Oxford)

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• 12th January 2026 - Jon Noel (University of Victoria)

  Less Than Half of All Oriented Paths Are Anti-Sidorenko

  
  

  An oriented graph $H$ is "tournament anti-Sidorenko" if the number of homomorphisms from $H$ to any tournament $T$ is at most $(1/2)^{|A(H)|}|V(T)|^{|V(H)|}$. He, Mani, Nie and Tung conjectured that a random orientation of a long path is tournament anti-Sidorenko with probability at least $1/2 - o(1)$. We disprove their conjecture by proving that the probability of this event is at most $0.445 +o(1)$. Our proof uses a Central Limit Theorem of Janson and the Cramér–Wold Device from probability theory together with ideas from combinatorial limits. Joint work with Hao Chen, Felix Christian Clemen and Anastasiia Sharfenberg.