Workshop

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We will discuss a few simple (to state) questions about induced matchings, together with some old and new connections between these questions and various topics at the intersection of extremal combinatorics, discrete geometry and number theory. 

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Lots of problems in combinatorics and analysis are connected to upper bounds for incidences: given a set of points and tubes, how much can they intersect? On the other hand, lower bounds for incidences have not been studied much. In this vein, we prove that if you choose `n’ points in the unit square and a line through each point, then there is a nontrivial point-line pair with distance <= n^{-2/3+o(1)}. It quickly follows that in any set of `n’ points in the unit square some three form a triangle of area <= n^{-7/6+o(1)}, a new bound for this problem. The main work is proving a more general incidence lower bound result under a new regularity condition. Joint work with Cosmin Pohoata and Dmitrii Zakharov.

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Elekes and Rónyai proved that if P(x,y) is a polynomial, then either P has a very special structure (for example P(x,y) = x+y or P(x,y) = xy), or else for every pair of sets A, B \subset R of size n, P(A x B) has cardinality substantially larger than n. This result was generalized by Elekes and Szabó, who proved that if Z \subset R^3 is an algebraic variety, then either Z has a very special structure (for example Z is a plane), or else for every triples of sets A, B, C \subset R of size n, Z \cap (A x B x C) has cardinality substantially smaller than n^2.

The Elekes-Rónyai and Elekes-Szabó theorems have applications to extremal questions in incidence geometry. There have been several generalizations of the above results, and several quantitative improvements that have strengthened the exponent in the Elekes-Szabó theorem from n^2 to n^{2-c} for various explicit values of c > 0 (and similarly for the Elekes-Rónyai theorem). I will discuss some ideas that give further quantitative improvements for this problem, leading to the exponent c = 2/7. This is joint work with Jozsef Solymosi.

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The Guth--Katz polynomial partitioning was first introduced in their ground-breaking result on the Erdős distinct distances problem, and it has found many applications in incidence geometry since then. In this talk, I will discuss two recent works that apply the tool. The first is a joint work with Gabriel Currier and József Solymosi, which shows that near optimizers of the Szemerédi–Trotter theorem must be "rigid" in some sense. The second is a joint work with Jonathan Tidor, which concerns hypergraphs that are called semialgraic. A semialgebraic hypergraph is a hypergraph with vertices being points in some Euclidean space and edges defined by some semialgebraic relations. I will talk about how polynomial partitioning is useful to prove a strong regularity lemma and a Zarankiewciz-type result for semialgebraic hypergraphs.

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Given a real number λ and a finite set A of real numbers, how small can the size of the sum of dilate A + λ.A be in terms of |A|? If λ is transcendental, then |A + λ.A| grows superlinearly in |A|, whereas if λ is algebraic, then |A + λ.A| only grows linearly in |A|. There have been several works in recent years to prove optimal linear bounds in the algebraic case, but tight bounds were only known when λ is an algebraic integer or of the form (p/q)^{1/d}.

In this talk, we prove tight bounds for sums of arbitrarily many algebraic dilates |A + λ1.A + ... + λk.A|. We will discuss the main tools used in the proof, which include a Frieman-type structure theorem for sets with small sums of dilates, and a high-dimensional notion of density which we call "lattice density". Joint work with David Conlon.

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I'll speak about joint work with Rachel Greenfeld and Marina Iliopoulou in which we address some classical questions concerning the size and structure of integer distance sets. A subset of the Euclidean plane is said to be an integer distance set if the distance between any pair of points in the set is an integer. Our main result is that any integer distance set in the plane has all but a very small number of points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any non-collinear integer distance set of size n and a strong upper bound on the size of any integer distance set in [-N,N]^2 with no three points on a line and no four points on a circle.

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It is a fundamental question in Ramsey theory to determine the smallest possible independence number of an H-free hypergraph on n vertices. In the case of graphs, the problem was famously solved for H=K3 by Kim and for H=K4 (up to a logarithmic factor) by Mattheus-Verstraete in 2023. Even C4 and K5 remain wide open. We study the problem for 3-uniform hypergraphs and conjecture a full classification: the minimum independence number is poly(n) if and only if H is contained in the iterated blowup of the single-edge hypergraph. We prove this conjecture for all H with at most two tightly connected components. Based on joint work with Conlon, Fox, Gunby, Mubayi, Suk, Verstraete, and Yu.

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We consider the question of determining the number of pentagons in a linear triple system and show some connections to number theory, graph theory and geometry. This is joint work with Jozsef Solymosi.