arXiv subjects

Matthew Kwan

Publications and source records attributed to Matthew Kwan.

At least 19 recordsLinked to original sources

arXiv2026

No-$(k+1)$-in-line problem for large constant $k$

How many points can be placed in an $n\times n$ grid so that every (affine) line contains at most $k$ points? We prove that for $n \geqslant k \geqslant 10^{37}$ the maximum number of points is exactly $kn$. Our proof builds on the recent work of Kovács, Nagy, and Szabó (who proved an analogous result when $k$ is at least about $\sqrt{n \log n}$), incorporating ideas of Jain and Pham. Using the same approach, we also obtain new bounds for higher-dimensional extensions of this problem.

math.CO↗

arXiv2026

No-$(k+1)$-in-line problem for $k \geqslant 3$

What is the maximum number of points one can place in an $n \times n$ grid such that every Euclidean line contains at most $k$ points? For $k = 2$, this is the notorious no-three-in-line problem of Dudeney. In this paper, we resolve this problem for all other $k$ (and sufficiently large $n$). Namely, for $k \geqslant 3$ and sufficiently large $n$, we show that this maximum is exactly $kn$. To prove this, our key observation is that in the regime $k \geqslant 3$, the problem is dominated in a certain statistical sense by the influence of a small number of "heavy" lines with many grid points. We apply a result of Ehard-Glock-Joos on pseudorandom hypergraph matchings to construct a set of size $kn - o(n)$ with at most $k$ points on each heavy line, and then a crude deletion argument yields a no-$(k+1)$-in-line set of nearly the same size. Finally, we use a randomised switching procedure to complete the construction (building upon ideas of Simkin and Luria). Using similar ideas, we also address the no-four-on-a-circle problem of Erdős and Purdy. Namely, we prove the existence of a set of $2n - o(n)$ points in the $n \times n$ grid such that no four of these points lie on a circle or a line, improving on the previous construction of size $n - o(n)$ due to Dong and Xu.

math.CO↗

arXiv2026

Entangled states are typically incomparable

Consider a bipartite quantum system, where Alice and Bob jointly possess a pure state $|ψ\rangle$. Using local quantum operations on their respective subsystems, and unlimited classical communication, Alice and Bob may be able to transform $|ψ\rangle$ into another state $|ϕ\rangle$. Famously, Nielsen's theorem [Phys. Rev. Lett., 1999] provides a necessary and sufficient algebraic criterion for such a transformation to be possible (namely, the local spectrum of $|ϕ\rangle$ should majorise the local spectrum of $|ψ\rangle$). In the paper where Nielsen proved this theorem, he conjectured that in the limit of large dimensionality, for almost all pairs of states $|ψ\rangle, |ϕ\rangle$ (according to the natural unitary invariant measure) such a transformation is not possible. That is to say, typical pairs of quantum states $|ψ\rangle, |ϕ\rangle$ are entangled in fundamentally different ways, that cannot be converted to each other via local operations and classical communication. Via Nielsen's theorem, this conjecture can be equivalently stated as a conjecture about majorisation of spectra of random matrices from the so-called trace-normalised complex Wishart-Laguerre ensemble. Concretely, let $X$ and $Y$ be independent $n \times m$ random matrices whose entries are i.i.d. standard complex Gaussians; then Nielsen's conjecture says that the probability that the spectrum of $X X^\dagger / \operatorname{tr}(X X^\dagger)$ majorises the spectrum of $Y Y^\dagger / \operatorname{tr}(Y Y^\dagger)$ tends to zero as both $n$ and $m$ grow large. We prove this conjecture, and we also confirm some related predictions of Cunden, Facchi, Florio and Gramegna [J. Phys. A., 2020; Phys. Rev. A., 2021].

quant-ph↗

arXiv2026

Permanents of random matrices over finite fields

Fix a finite field $\mathbb F_q$ and let $A\in \mathbb F_q^{n\times n}$ be a uniformly random $n\times n$ matrix over $\mathbb F_q$. The asymptotic distribution of the determinant $\det(A)$ is well-understood, but the asymptotic distribution of the permanent $\operatorname{per}(A)$ is still something of a mystery. In this paper we make a first step in this direction, proving that $\operatorname{per}(A)$ is significantly more uniform than $\det(A)$.

math.CO↗

arXiv2025

Counting Perfect Matchings In Dirac Hypergraphs

One of the foundational theorems of extremal graph theory is Dirac's theorem, which says that if an n-vertex graph G has minimum degree at least n/2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by Sárkozy, Selkow and Szemerédi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler and Kahn's result to perfect matchings in hypergraphs. For positive integers d < k, and for n divisible by k, let $m_{d}(k,n)$ be the minimum d-degree that ensures the existence of a perfect matching in an n-vertex k-uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of $m_{d}(k,n)$, but we are nonetheless able to prove an analogue of the Cuckler-Kahn theorem, showing that if an n-vertex k-uniform hypergraph G has minimum d-degree at least $(1+γ)m_{d}(k,n)$ (for any constant $γ>0$), then the number of perfect matchings in G is controlled by an entropy-like parameter of G. This strengthens cruder estimates arising from work of Kang-Kelly-Kühn-Osthus-Pfenninger and Pham-Sah-Sawhney-Simkin.

math.CO↗

arXiv2025

Edge-statistics beyond $1/e$

For integers $k$ and $\ell$, let $\operatorname{ind}(k, \ell)$ be the maximum proportion of $k$-vertex subsets of a large graph that induce exactly $\ell$ edges. The edge-statistics theorem (conjectured by Alon-Hefetz-Krivelevich-Tyomkyn, and proved by Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Trujić) asserts that, for $k \to \infty$ and $0 < \ell <\binom{k}{2}$, one has $\operatorname{ind}(k, \ell) \le 1/e + o(1)$. We investigate the ''stability'' of this problem: how can one improve this bound under additional assumptions on $\ell$? In particular, the edge-statistics theorem is tight when $\ell\in \{1,k-1,\binom{k}{2}-(k-1),\binom{k}{2}-1\}$; we show that for all other $\ell$, one can replace $1/e$ with a strictly smaller constant. This extends an analogous result of Ueltzen in the setting of graph inducibility. We also obtain a much stronger (and essentially optimal) upper bound on $\operatorname{ind}(k, \ell)$ when $\ell$ is far from a multiple of $k$, refining and extending previous bounds due to Fox and Sauermann.

math.CO↗

arXiv2025

Smoothed analysis for graph isomorphism

There is no known polynomial-time algorithm for graph isomorphism testing, but elementary combinatorial "refinement" algorithms seem to be very efficient in practice. Some philosophical justification is provided by a classical theorem of Babai, Erdős and Selkow: an extremely simple polynomial-time combinatorial algorithm (variously known as "naïve refinement", "naïve vertex classification", "colour refinement" or the "1-dimensional Weisfeiler-Leman algorithm") yields a so-called canonical labelling scheme for "almost all graphs". More precisely, for a typical outcome of a random graph $G(n,1/2)$, this simple combinatorial algorithm assigns labels to vertices in a way that easily permits isomorphism-testing against any other graph. We improve the Babai-Erdős-Selkow theorem in two directions. First, we consider randomly perturbed graphs, in accordance with the smoothed analysis philosophy of Spielman and Teng: for any graph $G$, naïve refinement becomes effective after a tiny random perturbation to $G$ (specifically, the addition and removal of $O(n\log n)$ random edges). Actually, with a twist on naïve refinement, we show that $O(n)$ random additions and removals suffice. These results significantly improve on previous work of Gaudio-Rácz-Sridhar, and are in certain senses best-possible. Second, we complete a long line of research on canonical labelling of random graphs: for any $p$ (possibly depending on $n$), we prove that a random graph $G(n,p)$ can typically be canonically labelled in polynomial time. This is most interesting in the extremely sparse regime where $p$ has order of magnitude $c/n$; denser regimes were previously handled by Bollobás, Czajka-Pandurangan, and Linial-Mosheiff. Our proof also provides a description of the automorphism group of a typical outcome of $G(n,p_n)$ (slightly correcting a prediction of Linial-Mosheiff).

math.CO↗

arXiv2025

On random matrices with large corank

Let $1\le k\le n$ and $M$ be a random $n\times n$ matrix with independent uniformly random $\{\pm 1\}$-entries. We show that there exists an absolute constant $c > 0$ such that \[\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).\]

math.PR↗

arXiv2025

Exponential anticoncentration of the permanent

Let $A\in\mathbb{R}^{n\times n}$ be a random matrix with independent entries, and suppose that the entries are "uniformly anticoncentrated" in the sense that there is a constant $\varepsilon>0$ such that each entry $a_{ij}$ satisfies $\sup_{z}\Pr[a_{ij}=z]\le1-\varepsilon$ (for example, $A$ could be a uniformly random $n\times n$ matrix with $\pm1$ entries). Significantly improving previous bounds of Tao and Vu, we prove that the permanent of $A$ is exponentially anticoncentrated: there is $c_{\varepsilon}>0$ such that $\sup_{z}\Pr[\operatorname{per}(A)=z]\le\exp(-c_{\varepsilon}n)$. Our proof also works for the determinant, giving an alternative proof of a classical theorem of Kahn, Komlós and Szemerédi. As a consequence, we see that there are at least exponentially many different permanents of $n\times n$ matrices with $\pm1$ entries, resolving a problem of Ingram and Razborov.

math.PR↗

arXiv2025

Parities in random Latin squares

In a Latin square, every row can be interpreted as a permutation, and therefore has a parity (even or odd). We prove that in a uniformly random $n\times n$ Latin square, the $n$ row parities are very well approximated by a sequence of $n$ independent unbiased coin flips: for example, the total variation error of this approximation tends to zero as $n\to\infty$. This resolves a conjecture of Cameron. In fact, we prove a generalisation of Cameron's conjecture for the joint distribution of the row parities, column parities and symbol parities (the latter are defined by the symmetry between rows, columns and symbols of a Latin square). Along the way, we introduce several general techniques for the study of random Latin squares, including a new re-randomisation technique via `stable intercalate switchings', and a new approximation theorem comparing random Latin squares with a certain independent model.

math.PR↗

arXiv2025

The edge-statistics conjecture for hypergraphs

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be exactly 1 (by taking $G$ to be empty or complete), but for all other values of $\ell$, one might suspect that this fraction is always significantly smaller than 1. In this paper we prove an essentially optimal result along these lines: if $\ell$ is not 0 or $\binom{k}{r}$, then this fraction is at most $(1/e) + \varepsilon$, assuming $k$ is sufficiently large in terms of $r$ and $\varepsilon>0$, and $G$ is sufficiently large in terms of $k$. Previously, this was only known for a very limited range of values of $r,k,\ell$ (due to Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Trujić). Our result answers a question of Alon-Hefetz-Krivelevich-Tyomkyn, who suggested this as a hypergraph generalisation of their "edge-statistics conjecture". We also prove a much stronger bound when $\ell$ is far from 0 and $\binom{k}{r}$.

math.CO↗

arXiv2025

Geometric Littlewood-Offord problems via lattice point counting

Consider nonzero vectors $a_{1},\dots,a_{n}\in\mathbb{C}^{k}$, independent Rademacher random variables $ξ_{1},\dots,ξ_{n}$, and a set $S\subseteq\mathbb{C}^{k}$. What upper bounds can we prove on the probability that the random sum $ξ_{1}a_{1}+\dots+ξ_{n}a_{n}$ lies in $S$? We develop a general framework that allows us to reduce problems of this type to counting lattice points in $S$. We apply this framework with known results from diophantine geometry to prove various bounds when $S$ is a set of points in convex position, an algebraic variety, or a semialgebraic set. In particular, this resolves conjectures of Fox-Kwan-Spink and Kwan-Sauermann. We also obtain some corollaries for the polynomial Littlewood-Offord problem, for polynomials that have bounded Chow rank (i.e., can be written as a polynomial of a bounded number of linear forms). For example, one of our results confirms a conjecture of Nguyen and Vu in the special case of polynomials with bounded Chow rank: if a bounded-degree polynomial $F\in\mathbb{C}[x_{1},\dots,x_{n}]$ has bounded Chow rank and ''robustly depends on at least $b$ of its variables'', then $\mathbb{P}[F(ξ_{1},\dots,ξ_{n})=0]\le O(1/\sqrt{b})$. We also prove significantly stronger bounds when $F$ is ''robustly irreducible'', towards a conjecture of Costello.

math.CO↗

arXiv2025

Algebraic aspects of the polynomial Littlewood-Offord problem

Consider a degree-$d$ polynomial $f(ξ_1,\dots,ξ_n)$ of independent Rademacher random variables $ξ_1,\dots,ξ_n$. To what extent can $f(ξ_1,\dots,ξ_n)$ concentrate on a single point? This is the so-called polynomial Littlewood-Offord problem. A nearly optimal bound was proved by Meka, Nguyen and Vu: the point probabilities are always at most about $1/\sqrt n$, unless $f$ is "close to the zero polynomial" (having only $o(n^d)$ nonzero coefficients). In this paper we prove several results supporting the general philosophy that the Meka-Nguyen-Vu bound can be significantly improved unless $f$ is "close to a polynomial with special algebraic structure", drawing some comparisons to phenomena in analytic number theory. In particular, one of our results is a corrected version of a conjecture of Costello on multilinear forms (in an appendix with Ashwin Sah and Mehtaab Sawhney, we disprove Costello's original conjecture).

math.CO↗

arXiv2025

A central limit theorem for the matching number of a sparse random graph

In 1981, Karp and Sipser proved a law of large numbers for the matching number of a sparse Erdős-Rényi random graph, in an influential paper pioneering the so-called differential equation method for analysis of random graph processes. Strengthening this classical result, and answering a question of Aronson, Frieze and Pittel, we prove a central limit theorem in the same setting: the fluctuations in the matching number of a sparse random graph are asymptotically Gaussian. Our new contribution is to prove this central limit theorem in the subcritical and critical regimes, according to a celebrated algorithmic phase transition first observed by Karp and Sipser. Indeed, in the supercritical regime, a central limit theorem has recently been proved in the PhD thesis of Kreačić, using a stochastic generalisation of the differential equation method (comparing the so-called Karp-Sipser process to a system of stochastic differential equations). Our proof builds on these methods, and introduces new techniques to handle certain degeneracies present in the subcritical and critical cases. Curiously, our new techniques lead to a non-constructive result: we are able to characterise the fluctuations of the matching number around its mean, despite these fluctuations being much smaller than the error terms in our best estimates of the mean. We also prove a central limit theorem for the rank of the adjacency matrix of a sparse random graph.

math.CO↗

arXiv2025

Colouring random Hasse diagrams and box-Delaunay graphs

Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to axis-parallel boxes (where we put an edge between $u,v\in P$ whenever there is an axis-parallel box containing $u,v$ and no other points of $P$). In each of these two closely related settings, we show that the chromatic number of $G$ is typically $(\log n)^{d-1+o(1)}$ and the independence number of $G$ is typically $n/(\log n)^{d-1+o(1)}$. When $d=2$, we obtain bounds that are sharp up to constant factors: the chromatic number is typically of order $\log n/\log\log n$ and the independence number is typically of order $n\log\log n/\log n$. These results extend and sharpen previous bounds by Chen, Pach, Szegedy and Tardos. In addition, they provide new bounds on the largest possible chromatic number (and lowest possible independence number) of a $d$-dimensional box-Delaunay graph or Hasse diagram, in particular resolving a conjecture of Tomon.

math.CO↗

arXiv2024

Books, Hallways and Social Butterflies: A Note on Sliding Block Puzzles

Recall the classical 15-puzzle, consisting of 15 sliding blocks in a $4\times 4$ grid. Famously, the configuration space of this puzzle consists of two connected components, corresponding to the odd and even permutations of the symmetric group $S_{15}$. In 1974, Wilson generalised sliding block puzzles beyond the $4\times 4$ grid to arbitrary graphs (considering $n-1$ sliding blocks on a graph with $n$ vertices), and characterised the graphs for which the corresponding configuration space is connected. In this work, we extend Wilson's characterisation to sliding block puzzles with an arbitrary number of blocks (potentially leaving more than one empty vertex). For any graph, we determine how many empty vertices are necessary to connect the corresponding configuration space, and more generally we provide an algorithm to determine whether any two configurations are connected. Our work may also be interpreted within the framework of "Friends and Strangers graphs", where empty vertices correspond to "social butterflies" and sliding blocks to "asocial" people.

math.CO↗

arXiv2024

Partitioning problems via random processes

There are a number of well-known problems and conjectures about partitioning graphs to satisfy local constraints. For example, the majority colouring conjecture of Kreutzer, Oum, Seymour, van der Zypen and Wood states that every directed graph has a 3-colouring such that for every vertex $v$, at most half of the out-neighbours of $v$ have the same colour as $v$. As another example, the internal partition conjecture, due to DeVos and to Ban and Linial, states that for every $d$, all but finitely many $d$-regular graphs have a partition into two nonempty parts such that for every vertex $v$, at least half of the neighbours of $v$ lie in the same part as $v$. We prove several results in this spirit: in particular, two of our results are that the majority colouring conjecture holds for Erdős-Rényi random directed graphs (of any density), and that the internal partition conjecture holds if we permit a tiny number of "exceptional vertices". Our proofs involve a variety of techniques, including several different methods to analyse random recolouring processes. One highlight is a "personality-changing" scheme: we "forget" certain information based on the state of a Markov chain, giving us more independence to work with.

math.CO↗

arXiv2024

Exponentially many graphs are determined by their spectrum

As a discrete analogue of Kac's celebrated question on "hearing the shape of a drum", and towards a practical graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by their spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers, is that "almost all graphs are determined by their spectrum", meaning that the fraction of unlabelled $n$-vertex graphs which are determined by their spectrum converges to $1$ as $n\to\infty$. In this paper we make a step towards this conjecture, showing that there are exponentially many $n$-vertex graphs which are determined by their spectrum. This improves on previous bounds (of shape $e^{c\sqrt{n}}$). We also propose a number of further directions of research.

math.CO↗