 # Matthew Kwan

Office: 382-J (second floor)
Department of Mathematics
450 Jane Stanford Way
Building 380
Stanford, CA 94305-2125
United States

Hi! Since September 2018, I'm a postdoc (Szegő Assistant Professor) in the mathematics department of Stanford University, mentored by Jacob Fox. Before coming to Stanford, my doctoral studies were at ETH Zurich under the guidance of Benny Sudakov, and my undergraduate studies were at UNSW (Sydney), where my adviser was Catherine Greenhill. You can find my CV here.

In Autumn 2019 I am teaching MATH 107: Graph Theory (course website).

## Selected research papers

(A complete list of papers is available here)
• Almost all Steiner triple systems are almost resolvable (with Asaf Ferber). Submitted.

Almost all Steiner triple systems have perfect matchings. Submitted.

We prove a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process. In a first paper, we use this theorem to show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (in fact, we show that almost all such Steiner triple systems have essentially the maximum possible number of perfect matchings). In a second paper, we show that almost all such Steiner triple systems in fact admit a decomposition of almost all their triples into disjoint perfect matchings. That is, almost all Steiner triple systems are almost resolvable.

• Halfway to Rota's basis conjecture (with Matija Bucic, Alexey Pokrovskiy and Benny Sudakov). Submitted.

Given n bases B1,...,Bn in an n-dimensional vector space V, a transversal basis is a basis of V containing exactly one element from each Bi. Rota's basis conjecture posits that it is always possible to find n disjoint transversal bases. In this paper we prove the partial result that one can always find (1/2 − o(1)) n disjoint transversal bases, improving on the previous record of Ω(n/log n). Our result generalises to the setting of matroids.

• Proof of a conjecture on induced subgraphs of Ramsey graphs (with Benny Sudakov). Transactions of the American Mathematical Society 372 (2019), 5571-5594.

Ramsey graphs induce subgraphs of quadratically many sizes (with Benny Sudakov). International Mathematics Research Notices, to appear.

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is a line of research towards showing that in fact all Ramsey graphs must obey certain “richness” properties characteristic of random graphs. In these papers we prove two conjectures along these lines. First, we prove that for any fixed C, every n-vertex C-Ramsey graph induces subgraphs of Θ(n2) different sizes. This resolves a conjecture of Narayanan, Sahasrabudhe and Tomon, motivated by an old problem of Erdős and McKay. Second, we prove a conjecture of Erdős, Faudree and Sós that in any n-vertex C-Ramsey graph, there are Ω(n5/2) induced subgraphs, no pair of which have the same numbers of edges and vertices.

• Anticoncentration for subgraph statistics (with Benny Sudakov and Tuan Tran). Journal of the London Mathematical Society 99.3 (2019), 757-777.

Consider integers k, l such that 0 ≤ l ≤ (k choose 2). Given a large graph G, what is the fraction of k-vertex subsets of G which span exactly l edges? When l is zero or (k choose 2), this fraction can be exactly 1. On the other hand, with Ramsey's theorem in mind, if l is far from these extreme values we might expect that this fraction must always be substantially smaller than 1. We prove an almost-best-possible theorem to this effect, improving on results of Alon, Hefetz, Krivelevich and Tyomkyn. We also make some first steps towards some analogous questions for hypergraphs. Our proofs take a probabilistic point of view, and involve polynomial anticoncentration inequalities, hypercontractivity, and a coupling trick for random variables defined on a “slice” of the Boolean hypercube.

• Resilience for the Littlewood-Offord Problem (with Afonso Bandeira and Asaf Ferber). Advances in Mathematics 319 (2017), 292-312.

Fix a sequence of nonzero real numbers a = (a1,...,an), consider a random ±1 sequence ξ = (ξ1,...,ξn), and let X = a1ξ1+...+anξn. The Erdős–Littlewood–Offord theorem shows that, regardless of a, for any x the event X = x is unlikely (that is, X is anti-concentrated). In this paper, motivated by some questions about random matrices, we study the “resilience” of this anti-concentration. For a given x, how many coordinates of ξ can we allow an adversary to change before they can force X = x? The answer is quite surprising, and its proof involves an interesting connection to combinatorial number theory.