Schedule

For a subset A of an additive group G, a Schur triple in A is a triple of the form (a,b,a+b) ∈ A3. Denote by r(A) the number of Schur triples of A; the behaviour of r(A) as A ranges over subsets of a group G has been studied by various authors. When r(A)=0, A is sum-free. The question of minimum and maximum r(A) for A of fixed size in ℤp was resolved by Huczynska, Mullen and Yucas (2009) and independently by Samotij and Sudakov (2016). Chervak, Pikhurko and Staden (2019) addressed generalized versions of the Schur triple question for (k+1)-tuples in ℤp. In this talk, I will explore this landscape and present recent work (with Jonathan Jedwab and Laura Johnson) on the generalization of the Schur triple problem to triples (a,b,a+b) ∈ A × B × B, where A,B ⊆ ℤp. Denote by r(A,B,B) the number of triples of this form; we obtain a precise description of its full spectrum of values and show constructively that each value in this spectrum can be realised when B is an interval of consecutive elements in ℤp.

The Turán density t(s,r) is the asymptotically smallest edge density of an r-graph for which every set of s vertices contains at least one edge. The question of estimating this function received a lot of attention since it was first raised by Turán in 1941. A trivial lower bound is t(s,r) ≥ 1/(s choose s−r). In the early 1990s, de Caen conjectured that t(r+1,r) grows faster than O(1/r) and offered 500 Canadian dollars for resolving this question.

I will give an overview of this area and present a construction disproving this conjecture by showing more strongly that for every integer R there is C such that t(r+R,r) ≤ C/(r+R choose R), that is, the trivial lower bound is tight for every fixed R up to a multiplicative constant C=C(R).

The celebrated Szemerédi-Trotter theorem states that the maximum number of incidences between n points and n lines in the plane is O(n4/3), which is asymptotically tight.

Solymosi conjectured that this bound drops to o(n4/3) if we exclude subconfigurations isomorphic to any fixed point-line configuration. We describe a construction disproving this conjecture. On the other hand, we prove new upper bounds on the number of incidences for configurations that avoid certain subconfigurations. Joint work with Martin Balko.

We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated (p,q)-theorem in discrete geometry. In particular, for graphs with bounded clique number and certain natural density condition, we prove a (p,q)-theorem for the dual of its maximal independent sets hypergraph. Our result strengthens those of Thomassen and Nikiforov on the chromatic threshold of cliques. We further show that the graphs under study in fact have `bounded complexity’ in the sense that they are blow-ups of constant size graphs with the same clique number, strengthening the results of Luczak and Goddard-Lyle on homomorphism threshold of cliques and improving the best known quantitative result of Oberkampf and Schacht. Our result unravels the cause underpinning such blow-up phenomenon, showing that rather than the minimum degree condition usually considered in the literature, the decisive factor is a density condition on co-neighbourhoods.

Joint work with Hong Liu, Chong Shangguan and Zixiang Xu.

Given an n-vertex graph G, let hom(G) denote the size of a largest homogeneous set in G and let f(G) denote the maximal number of distinct degrees appearing in an induced subgraph of G. The relationship between these parameters has been well studied by several researchers over the last 40 years, beginning with Erdős, Faudree and Sós in the Ramsey regime when hom(G) = O(log n).

In this talk I hope to discuss a recent theorem which provides the complete extremal relationship between these parameters (asymptotically), showing that any n-vertex graph G satisfies

max ( f(G) ⋅ hom(G), f(G)3/2 ⋅ hom(G)1/2 ) ≥ n1-o(1).

This relationship is tight (up to the n-o(1) term) for all possible values of hom(G), from order log n to n, as demonstrated by appropriately generated Erdős-Rényi random graphs.

Joint work with Laurenţiu Ploscaru.