Hi! Since September 2018, I'm a postdoc (Szegö 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.
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. See also our companion note adapting our methods to the setting of a related conjecture due to Kahn.
We show that for any n divisible by 3, almost all order-n Steiner triple systems have a perfect matching (also known as a parallel class). In fact, we prove a general upper bound on the number of perfect matchings in a Steiner triple system and show that almost all Steiner triple systems essentially attain this maximum. We accomplish this via a general theorem comparing a uniformly random Steiner triple system to the outcome of the triangle removal process, which we hope will be useful for other problems.
An r-cut of a k-uniform hypergraph (k-graph) H is a partition of the vertex set into r parts, and the size of such a cut is the number of edges which have a vertex from every part. The max-r-cut of H is the maximum size of an r-cut of H. We prove some new bounds on the max-r-cut of a k-graph, for fixed r ≤ k, above the trivial “average” bound obtainable from a uniformly random cut. In particular, in contrast to the situation for max-cut in graphs and max-2-cut in 3-graphs, we show that if k ≥ 4 or r ≥ 3 then the worst-case behaviour is not governed by the standard deviation of a uniformly random cut.
Consider integers k, l such that 0 ≤ l ≤ (k choose ). 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 ), 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.
Proof of a conjecture on induced subgraphs of Ramsey graphs (with Benny Sudakov). Transactions of the American Mathematical Society, to appear.
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. This improves on earlier results due to Alon, Balogh, Kostochka and Samotij.
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 is an adversary typically allowed to change without making X = x? The answer is quite surprising, and its proof involves an interesting connection to combinatorial number theory.