Gruppe 4

Department Mathematik

HG J 14.3

Rämistrasse 101

8092 Zürich

Switzerland

Since September 2014, I'm a doctoral student in mathematics at ETH Zurich under the guidance of Benny Sudakov. Before coming to ETH, my undergraduate studies were at UNSW (Sydney), where my adviser was Catherine Greenhill.

This semester I'm teaching graph theory. Here is the course homepage.

- Anticoncentration for subgraph statistics (with Benny Sudakov and Tuan Tran). Submitted.
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. - Almost all Steiner triple systems have perfect matchings. Submitted.
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. - Hypergraph cuts above the average (with David Conlon, Jacob Fox and Benny Sudakov). Submitted.
An

of a*r*-cut*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-*of*r*-cut*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. Proof of a conjecture on induced subgraphs of Ramsey graphs (with Benny Sudakov). Submitted.

Ramsey graphs induce subgraphs of quadratically many sizes (with Benny Sudakov).

*International Mathematics Research Notices*, to appear.An

*n*-vertex graph is calledif it has no clique or independent set of size*C*-Ramsey*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(*n*^{2}) 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(*n*^{5/2}) 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.- 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***a*_{1},...,*a*), consider a random ±1 sequence_{n}= (**ξ***ξ*_{1},...,*ξ*), and let_{n}*X*=*a*_{1}*ξ*_{1}+...+*a*. The Erdős-Littlewood-Offord theorem shows that, regardless of_{n}ξ_{n}, for any**a***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 ofis an adversary typically allowed to change without making**ξ***X = x*? The answer is quite surprising, and its proof involves an interesting connection to combinatorial number theory.