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

• 2nd March 2026 - Laura Johnson (University of Bristol)

  Additive triples in groups of odd prime order

  
  

  For nontrivial subsets $A$ and $B$ of $G$, we define an additive triple to be a triple of the form $(a, b, a + b) \in A \times B \times B$. We say that $r(A, B, B)$ is the number of additive triples for two subsets $A$ and $B$ of $G$ with cardinality $s$ and $t$, respectively. For fixed values of $s$ and $t$, it is an interesting problem to consider the spectrum of $r(A, B, B)$ values.

  In the special case where $A = B$, the additive triples are equivalent to Schur triples. In the group $\mathbf Z_p$, the spectrum of $r(A, A, A)$ values for a subset $A$ of $G$ with fixed cardinality $s \leqslant p$ have been well studied in the literature. However, it is known that for certain values of $p$ and $s$, not every value of $r(A, A, A)$ between the upper and lower bound in attainable.

  In this talk, we will look at the case where $A$ and $B$ can be distinct. In this case, we can use Pollard’s Theorem, which is a generalisation of the Cauchy–Davenport Theorem, to establish an upper and lower bound for the number $r(A, B, B)$ of additive triples of cardinality $s$ and $t$. We demonstrate that in this case, it is possible to attain every value of $r(A, B, B)$ between the upper and lower bound.

  This talk is based on joint work with Dr. Sophie Huczysnka (University of St. Andrews) and Professor Jonathan Jedwab (Simon Fraser University).