Discrete Mathematics Group

Tony Huynh, Aharoni's rainbow cycle conjecture holds up to an additive constant
2024-05-07 16:30 - 17:30 (Asia/Seoul) | Room B332, IBS (기초과학연구원)
In 2017, Aharoni proposed the following generalization of the Caccetta-Häggkvist conjecture for digraphs. If G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most ⌈n/r⌉.
In this talk, we prove that Aharoni’s conjecture holds up to an additive constant. Specifically, we show that for each fixed r, there exists a constant c such that if G is a simple n-vertex edge-colored graph with n color classes of size at least r, then G contains a rainbow cycle of length at most n/r+c.
This is joint work with Patrick Hompe.

Niloufar Fuladi, Cross-cap drawings and signed reversal distance
2024-05-14 16:30 - 17:30
A cross-cap drawing of a graph G is a drawing on the sphere with g distinct points, called cross-caps, such that the drawing is an embedding except at the cross-caps, where edges cross properly. A cross-cap drawing of a graph G with g cross-caps can be used to represent an embedding of G on a non-orientable surface of genus g. Mohar conjectured that any triangulation of a non-orientable surface of genus g admits a cross-cap drawing with g cross-caps in which each edge of the triangulation enters each cross-cap at most once. Motivated by Mohar’s conjecture, Schaefer and Stefankovic provided an algorithm that computes a cross-cap drawing with a minimal number of cross-caps for a graph G such that each edge of the graph enters each cross-cap at most twice. In this talk, I will first outline a connection between cross-cap drawings and an algorithm coming from computational biology to compute the signed reversal distance between two permutations. This connection will then be leveraged to answer two computational problems on graphs embedded on surfaces.
First, I show how to compute a “short” canonical decomposition for a non-orientable surface with a graph embedded on it. Such canonical decompositions were known for orientable surfaces, but the techniques used to compute them do not generalize to non-orientable surfaces due to their more complex nature. Second, I explain how to build a counter example to a stronger version of Mohar’s conjecture that is stated for pseudo-triangulations.
This is joint work with Alfredo Hubard and Arnaud de Mesmay.

Vadim Lozin, Graph problems and monotone classes
2024-05-21 16:30 - 17:30
Very little is known about critical properties of graphs in the hierarchy of monotone classes, i.e. classes closed under taking (not necessarily induced) subgraphs. We distinguish four important levels in this hierarchy and discuss possible new levels by focusing on the Hamiltonian cycle problem. In particular, we obtain a number of results for this problem on monotone classes.

Yongho Shin (신용호), Three-way online correlated selection
2024-05-28 16:30 - 17:30
Two-way online correlated selection (two-way OCS) is an online algorithm that, at each timestep, takes a pair of elements from the ground set and irrevocably chooses one of the two elements, while ensuring negative correlation in the algorithm’s choices. OCS was initially invented by Fahrbach, Huang, Tao, and Zadimoghaddam (FOCS 2020, JACM 2022) to break a natural long-standing barrier in edge-weighted online bipartite matching. They posed two open questions, one of which was the following: Can we obtain n-way OCS for $n >2$, in which the algorithm can be given $n >2$ elements to choose from at each timestep?
In this talk, we affirmatively answer this open question by presenting a three-way OCS which is simple to describe: it internally runs two instances of two-way OCS, one of which is fed with the output of the other. Contrast to its simple construction, we face a new challenge in analysis that the final output probability distribution of our three-way OCS is highly elusive since it requires the actual output distribution of two-way OCS. We show how we tackle this challenge by approximating the output distribution of two-way OCS by a flatter distribution serving as a safe surrogate.
This is joint work with Hyung-Chan An.

Jane Tan, Semi-strong colourings of hypergraphs
2024-06-04 16:30 - 17:30
A vertex colouring of a hypergraph is $c$-strong if every edge $e$ sees at least $min{c, |e|}$ distinct colours. Let $chi(t,c)$ denote the least number of colours needed so that every $t$-intersecting hypergraph has a $c$-strong colouring. In 2012, Blais, Weinstein and Yoshida introduced this parameter and initiated study on when $chi(t,c)$ is finite: they showed that $chi(t,c)$ is finite whenever $t geq c$ and unbounded when $tleq c-2$. The boundary case $chi(c-1, c)$ has remained elusive for some time: $chi(1,2)$ is known to be finite by an easy classical result, and $chi(2,3)$ was shown to be finite by Chung and independently by Colucci and Gyárfás in 2013. In this talk, we present some recent work with Kevin Hendrey, Freddie Illingworth and Nina Kamčev in which we fill in this gap by showing that $chi(c-1, c)$ is finite in general.

Maria Chudnovsky, Anticomplete subgraphs of large treewidth
2024-06-11 16:30 - 17:30
We will discuss recent progress on the topic of induced subgraphs and tree-decompositions. In particular this talk with focus on the proof of a conjecture of Hajebi that asserts that (if we exclude a few obvious counterexamples) for every integer t, every graph with large enough treewidth contains two anticomplete induced subgraphs each of treewidth at least t. This is joint work with Sepher Hajebi and Sophie Spirkl.

Semin Yoo (유세민), Paley-like quasi-random graphs arising from polynomials
2024-06-18 16:30 - 17:30
We provide new constructions of families of quasi-random graphs that behave like Paley graphs but are neither Cayley graphs nor Cayley sum graphs. These graphs give a unified perspective of studying various graphs defined by polynomials over finite fields, such as Paley graphs, Paley sum graphs, and graphs associated with Diophantine tuples and their generalizations from number theory. As an application, we provide new lower bounds on the clique number and independence number of general quasi-random graphs. In particular, we give a sufficient condition for the clique number of quasi-random graphs of order $n$ to be at least $(1-o(1))log_{3.008}n$. Such a condition applies to many classical quasi-random graphs, including Paley graphs and Paley sum graphs, as well as some new Paley-like graphs we construct. If time permits, we also discuss some problems of diophantine tuples arising from number theory, which is our original motivation.
This is joint work with Seoyoung Kim and Chi Hoi Yip.

Wonwoo Kang (강원우), Skein relations for punctured surfaces
2024-06-28 16:30 - 17:30
Since the introduction of cluster algebras by Fomin and Zelevinsky in 2002, there has been significant interest in cluster algebras of surface type. These algebras are particularly noteworthy due to their ability to construct various combinatorial structures like snake graphs, T-paths, and posets, which are useful for proving key structural properties such as positivity or the existence of bases. In this talk, we will begin by presenting a cluster expansion formula that integrates the work of Musiker, Schiffler, and Williams with contributions from Wilson, utilizing poset representatives for arcs on triangulated surfaces. Using these posets and the expansion formula as tools, we will demonstrate skein relations, which resolve intersections or incompatibilities between arcs. Topologically, a skein relation replaces intersecting arcs or arcs with self-intersections with two sets of arcs that avoid the intersection differently. Additionally, we will show that all skein relations on punctured surfaces include a term that is not divisible by any coefficient variable. Consequently, we will see that the bangles and bracelets form spanning sets and exhibit linear independence. This work is done in collaboration with Esther Banaian and Elizabeth Kelley.

Kisun Lee (이기선), Symmetric Tropical Rank 2 Matrices
2024-07-02 16:30 - 17:30
Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. We discuss that this tropicalization has a simplicial complex structure as the space of symmetric bicolored trees. As a result, we show that this space is shellable and delve into its matroidal structure. It is based on the joint work with May Cai and Josephine Yu.

Hyunwoo Lee (이현우), Random matchings in linear hypergraphs
2024-07-05 16:30 - 17:30
For a given hypergraph $H$ and a vertex $vin V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $vin V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998. In this paper, we disprove this conjecture for all $k geq 3.$ For infinitely many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $mathcal{P}(v_1
otin M) = 1 – frac{(1 + o_d(1))}{d^{k-2}}$ and $mathcal{P}(v_2
otin M) = frac{(1 + o_d(1))}{d+1}.$ The gap between $mathcal{P}(v_1
otin M)$ and $mathcal{P}(v_2
otin M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

Euiwoong Lee (이의웅), Parameterized Approximability of F-Deletion Problems
2024-07-30 16:30 - 17:30
For a family F of graphs, the F-Deletion Problem asks to remove the minimum number of vertices from a given graph G to ensure that G belongs to F. One of the most common ways to obtain an interesting family F is to fix another family H of graphs and let F be the set of graphs that do not contain any graph H as some notion of a subgraph, including (standard) subgraph, induced subgraph, and minor. This framework captures numerous basic graph problems, including Vertex Cover, Feedback Vertex Set, and Treewidth Deletion, and provides an interesting forum where ideas from approximation and parameterized algorithms influence each other. In this talk, I will give a brief survey on the state of the art on the F-Deletion Problems for the above three notions of subgraphs, and talk about a recent result on Weighted Bond Deletion.

Daniel Král', Matroid depth and width parameters
2024-08-06 16:30 - 17:30
Depth and width parameters of graphs, e.g., tree-width, path-width and tree-depth, play a crucial role in algorithmic and structural graph theory. These notions are of fundamental importance in the theory of graph minors, fixed parameter complexity and the theory of sparsity.
In this talk, we will survey structural and algorithmic results that concern width and depth parameters of matroids. We will particularly focus on matroid depth parameters and discuss the relation of the presented concepts to discrete optimization. As an application, we will present matroid based algorithms that uncover a hidden Dantzig-Wolfe-like structure of an input instance (if such structure is present) and transform instances of integer programming to equivalent ones, which are amenable to the existing tools in integer programming.
The most recent results presented in the talk are based on joint work with Marcin Briański, Jacob Cooper, Timothy F. N. Chan, Martin Koutecký, Ander Lamaison, Kristýna Pekárková and Felix Schröder.

Peter Nelson, Formalizing matroid theory in a proof assistant
2024-08-13 16:30 - 17:30
For the past few years, I’ve been working on formalizing proofs in matroid theory using the Lean proof assistant. This has led me to many interesting and unexpected places. I’ll talk about what formalization looks like in practice from the perspective of a combinatorialist.

Dillon Mayhew, Monadic second-order definability for gain-graphic matroids
2024-08-27 16:30 - 17:30
Every (finite) matroid consists of a (finite) set called the ground set, and a collection of distinguished subsets called the independent sets. A classic example arises when the ground set is a finite set of vectors from a vector space, and the independent subsets are exactly the subsets that are linearly independent. Any such matroid is said to be representable. We can think of a representable matroid as being a geometrical configuration where the points have been given coordinates from a field. Another important class arises when the points are given coordinates from a group. Such a class is said to be gain-graphic.
Monadic second-order logic is a natural language for matroid applications. In this language we are able to quantify only over subsets of the ground set. The importance of monadic second-order logic comes from its connections to the theory of computation, as exemplified by Courcelle’s Theorem. This theorem provides polynomial-time algorithms for recognising properties defined in monadic second-order logic (as long as we impose a bound on the structural complexity of the input objects). It is natural to ask which classes of matroids can be defined by sentences in monadic second-order logic. When the class consists of the matroids that are coordinatized by a field we have a complete answer to this question. When the class is coordinatized by a group the problem becomes much harder.
This talk will contain a brief introduction to matroids. Based on work with Sapir Ben-Shahar, Matt Conder, Daryl Funk, Angus Matthews, Mike Newman, and Gabriel Verret.

Amadeus Reinald, Oriented trees in $O(k sqrt{k})$-chromatic digraphs, a subquadratic bound for Burr's conjecture
2024-09-03 16:30 - 17:30
In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $frac{k^2}{2} – frac{k}{2} + 1$ by Addario-Berry et al.
In this talk, we give the first subquadratic bound for Burr’s conjecture, by showing that every directed graph with chromatic number $8sqrt{frac{2}{15}} k sqrt{k} + O(k)$ contains any oriented tree of order $k$. Moreover, we provide improved bounds of $sqrt{frac{4}{3}} k sqrt{k}+O(k)$ for arborescences, and $(b-1)(k-3)+3$ for paths on $b$ blocks, with $bge 2$.

Neal Bushaw, Edge-colored Extremal Problems
2024-09-06 16:30 - 17:30
An edge-colored graph $H$ is called rainbow if all of its edges are given distinct colors. An edge-colored graph $G$ is then called rainbow $H$-free when no copy of $H$ in $G$ is rainbow. With this, we define a graph $G$ to be rainbow $H$-saturated when there is some proper edge-coloring of $G$ which is rainbow $H$-free, but for every pair of non-adjacent vertices $x,yin V(G)$, the graph $G+xy$ formed by adding the edge $xy$ to $G$ cannot be given a rainbow $H$-free coloring. We think of these graphs as edge-maximal rainbow $H$-free graphs. (We note that here we make no restrictions on the colorings of $G+xy$ whatsoever, except that they are proper colorings. They may use any number of colors, and need not be extensions of the original rainbow $H$-free coloring of $G$.)

With this framework in place, we define the rainbow saturation number and rainbow extremal number to be the largest and smallest number of edges, respectively, among all $n$ vertex rainbow $H$-saturated graphs. The latter of these was defined by Keevash, Mubayi, Sudakov, and Verstraëte in 2007; the former was introduced by B., Johnston, and Rombach in 2019.

In this talk, we discuss recent progress on both the rainbow saturation numbers and rainbow extremal numbers. We also give several broad generalizations of these concepts and discuss many open problems. This talk contains joint work with Vic Bednar (Furman), Dan Johnston (Trinity College, CT), and Puck Rombach (Vermont).

Zhihan Jin (金之涵), The Helly number of Hamming balls and related problems
2024-09-10 16:30 - 17:30
We prove the following variant of Helly’s classical theorem for Hamming balls with a bounded radius. For $n > t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X| > 1$ and all $n > t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory. This is joint work with N. Alon and B. Sudakov.

Gábor Tardos, Extremal theory of 0-1 matrices
2024-09-24 16:30 - 17:30
We say that a 0-1 matrix A contains another such matrix (pattern) P if P can be obtained from a submatrix of A by possibly changing a few 1 entries to 0. The main question of this theory is to estimate the maximal number of 1 entries in an n by n 0-1 matrix NOT containing a given pattern P. This question has very close connections to Turan type extremal graph theory and also to the Devenport-Schinzel theory of sequences. Results in the extremal theory of 0-1 matrices proved useful in attacking problems in far away fields as combinatorial geometry and the analysis of algorithms.
This talk will concentrate on acyclic patterns and survey some old and recent results in the area and will also contain several open problems.

Mathias Schacht, Canonical colourings in random graphs
2024-10-08 16:30 - 17:30
Rödl and Ruciński established Ramsey’s theorem for random graphs. In particular, for fixed integers $r$, $ellgeq 2$ they showed that $n^{-frac{2}{ell+1}}$ is a threshold for the Ramsey property that every $r$-colouring of the edges of the binomial random graph $G(n,p)$ yields a monochromatic copy of $K_ell$.
We investigate how this result extends to arbitrary colourings of $G(n,p)$ with an unbounded number of colours. In this situation Erdős and Rado showed that canonically coloured copies of $K_ell$ can be ensured in the deterministic setting.
We transfer the Erdős-Rado theorem to the random environment and show that for $ellgeq 4$ both thresholds coincide. As a consequence the proof yields $K_{ell+1}$-free graphs $G$ for which every edge colouring yields a canonically coloured $K_ell$.
This is joint work with Nina Kamčev.

Kyeongsik Nam (남경식), Random walks on percolation
2024-10-15 16:30 - 17:30
In general, random walks on fractal graphs are expected to exhibit anomalous behaviors, for example heat kernel is significantly different from that in the case of lattices. Alexander and Orbach in 1982 conjectured that random walks on critical percolation, a prominent example of fractal graphs, exhibit mean field behavior; for instance, its spectral dimension is 4/3. In this talk, I will talk about this conjecture for a canonical dependent percolation model.

Colin Geniet, Permutations, patterns, and twin-width
2024-10-22 16:30 - 17:30
This talk will first introduce combinatorics on permutations and patterns, presenting the basic notions and some fundamental results: the Marcus-Tardos theorem which bounds the density of matrices avoiding a given pattern, and the Guillemot-Marx algorithm for pattern detection using the notion now known as twin-width.
I will then present a decomposition result: permutations avoiding a pattern factor into bounded products of separable permutations. This can be rephrased in terms of twin-width: permutation with bounded twin-width are build from a bounded product of permutations of twin-width 0. Comparable results on graph encodings follow from this factorisation.
This is joint work with Édouard Bonnet, Romain Bourneuf, and Stéphan Thomassé.

Felix Christian Clemen, Triangles in the Plane
2024-10-29 16:30 - 17:30
A classical problem in combinatorial geometry, posed by Erdős in 1946, asks to determine the maximum number of unit segments in a set of $n$ points in the plane. Since then a great variety of extremal problems in finite planar point sets have been studied. Here, we look at such questions concerning triangles. Among others we answer the following question asked by Erdős and Purdy almost 50 years ago: Given $n$ points in the plane, how many triangles can be approximate congruent to equilateral triangles?
For our proofs we use hypergraph Turán theory. This is joint work with Balogh and Dumitrescu.

Michał Pilipczuk, Monadic stability and monadic dependence
2024-11-05 16:30 - 17:30
We will give an overview of the recent attempts of building a structure theory for graphs centered around First-Order transductions: a notion of containment inspired by finite model theory. Particularly, we will speak about the notions of monadic dependence and monadic stability, their combinatorial characterizations, and the developments on the algorithmic front.

Karim Adiprasito, Ehrhart theory revisited: Algebraic aspects, unimodality and more
2024-11-12 16:30 - 17:30
Ehrhart theory is the study of lattice polytopes, specifically aimed at understanding how many lattice points are inside dilates of a given lattice polytope, and the study has a wide range of connections ranging from coloring graphs to mirror symmetry and representation theory. Recently, we introduced new algebraic tools to understand this theory, and resolve some classical conjectures. I will explain the combinatorial underpinnings behind two of the key techniques: Parseval identities for semigroup algebras, and the character algebra of a semigroup.

Eng Keat Hng, Graphon branching processes and fractional isomorphism
2024-11-26 16:30 - 17:30
In 2005, Bollobás, Janson and Riordan introduced and extensively studied a general model of inhomogeneous random graphs parametrised by graphons. In particular, they studied the emergence of a giant component in these inhomogeneous random graphs by relating them to a broad collection of inhomogeneous Galton-Watson branching processes.
Fractional isomorphism of finite graphs is an important and well-studied concept at the intersection of graph theory and combinatorial optimisation. It has many different characterizations that involve a range of very different and seemingly unrelated properties of graphs. Recently, Grebík and Rocha developed a theory of fractional isomorphism for graphons.
In our work, we characterise inhomogeneous random graphs that yield the same inhomogeneous Galton-Watson branching process (and hence have a similar component structure).
This is joint work with Jan Hladký and Anna Margarethe Limbach.

Yulai Ma, Pairwise disjoint perfect matchings in regular graphs
2024-12-03 16:30 - 17:30
An $r$-graph is an $r$-regular graph in which every odd set of vertices is connected to its complement by at least $r$ edges. A central question regarding $r$-graphs is determining the maximum number of pairwise disjoint perfect matchings they can contain. This talk explores how edge connectivity influences this parameter.
For ${0 leq lambda leq r}$, let $m(lambda,r)$ denote the maximum number $s$ such that every $lambda$-edge-connected $r$-graph contains $s$ pairwise disjoint perfect matchings. The values of $m(lambda,r)$ are known only in limited cases; for example, $m(3,3)=m(4,r)=1$, and $m(r,r) leq r-2$ for all $r
ot = 5$, with $m(r,r) leq r-3$ when $r$ is a multiple of $4$. In this talk, we present new upper bounds for $m(lambda,r)$ and examine connections between $m(5,5)$ and several well-known conjectures for cubic graphs.
This is joint work with Davide Mattiolo, Eckhard Steffen, and Isaak H. Wolf.

Jun Gao (高峻), Phase transition of degenerate Turán problems in p-norms
2024-12-13 16:30 - 17:30
For a positive real number $p$, the $p$-norm $|G|_p$ of a graph $G$ is the sum of the $p$-th powers of all vertex degrees. We study the maximum $p$-norm $mathrm{ex}_{p}(n,F)$ of $F$-free graphs on $n$ vertices, focusing on the case where $F$ is a bipartite graph. It is natural to conjecture that for every bipartite graph $F$, there exists a threshold $p_F$ such that for $p< p_{F}$, the order of $mathrm{ex}_{p}(n,F)$ is governed by pseudorandom constructions, while for $p > p_{F}$, it is governed by star-like constructions. We determine the exact value of $p_{F}$, under a mild assumption on the growth rate of $mathrm{ex}(n,F)$. Our results extend to $r$-uniform hypergraphs as well.
We also prove a general upper bound that is tight up to a $log n$ factor for $mathrm{ex}_{p}(n,F)$ when $p = p_{F}$.
We conjecture that this $log n$ factor is unnecessary and prove this conjecture for several classes of well-studied bipartite graphs, including one-side degree-bounded graphs and families of short even cycles.
This is a joint work with Xizhi Liu, Jie Ma and Oleg Pikhurko.

Joonkyung Lee (이준경), Counting homomorphisms in antiferromagnetic graphs via Lorentzian polynomials
2024-12-17 16:30 - 17:30
An edge-weighted graph $G$, possibly with loops, is said to be antiferromagnetic if it has nonnegative weights and at most one positive eigenvalue, counting multiplicities. The number of graph homomorphisms from a graph $H$ to an antiferromagnetic graph $G$ generalises various important parameters in graph theory, including the number of independent sets and proper vertex colourings.
We obtain a number of new homomorphism inequalities for antiferromagnetic target graphs $G$. In particular, we prove that, for any antiferromagnetic $G$,
$|mathrm{Hom}(K_d, G)|^{1/d} ≤ |mathrm{Hom}(K_{d,d} setminus M, G)|^{1/(2d)}$
holds, where $K_{d,d} setminus M$ denotes the complete bipartite graph $K_{d,d}$ minus a perfect matching $M$. This confirms a conjecture of Sah, Sawhney, Stoner and Zhao for complete graphs $K_d$. Our method uses the emerging theory of Lorentzian polynomials due to Brändén and Huh, which may be of independent interest.
Joint work with Jaeseong Oh and Jaehyeon Seo.

Zixiang Xu (徐子翔), Multilinear polynomial methods and stability results on set systems
2024-12-23 16:30 - 17:30
In 1966, Kleitman established that if ( |A triangle B| leq d ) for any ( A, B in mathcal{F} ), then ( |mathcal{F}| leq sum_{i=0}^{k} binom{n}{i} ) for ( d = 2k ), and ( |mathcal{F}| leq 2 sum_{i=0}^{k} binom{n-1}{i} ) for ( d = 2k+1 ). These upper bounds are attained by the radius-(k) Hamming ball ( mathcal{K}(n, k) := { F : F subseteq [n], |F| leq k } ) in the even case, and by the family ( mathcal{K}_y(n, k) := { F : F subseteq [n], |F setminus {y}| leq k } ) in the odd case. In 2017, Frankl provided a combinatorial proof of a stability result for Kleitman’s theorem, offering improved upper bounds for ( |mathcal{F}| ) when ( mathcal{F} ) is not the extremal structure.
In this talk, I will begin by demonstrating the application of multilinear polynomial methods in extremal set theory, highlighting some interesting techniques. I will then present an algebraic proof of the stability result for Kleitman’s theorem. Finally, I will discuss further applications and explore how to employ linear algebra methods more effectively and flexibly.
This talk is based on joint work with Jun Gao and Hong Liu.

Huy Tuan Pham, Random Cayley graphs and Additive combinatorics without groups
2025-01-03 16:30 - 17:30
A major goal of additive combinatorics is to understand the structures of subsets A of an abelian group G which has a small doubling K = |A+A|/|A|. Freiman’s celebrated theorem first provided a structural characterization of sets with small doubling over the integers, and subsequently Ruzsa in 1999 proved an analog for abelian groups with bounded exponent. Ruzsa further conjectured the correct quantitative dependence on the doubling K in his structural result, which has attracted several interesting developments over the next two decades. I will discuss a complete resolution of (a strengthening of) Ruzsa’s conjecture.
Surprisingly, our approach is crucially motivated by purely graph-theoretic insights, where we find that the group structure is superfluous and can be replaced by much more general combinatorial structures. Using this general approach, we also obtain combinatorial and nonabelian generalizations of classical results in additive combinatorics, and solve longstanding open problems around Cayley graphs and random Cayley graphs motivated by Ramsey theory, information theory and computer science.
Based on joint work with David Conlon, Jacob Fox and Liana Yepremyan.
