arXiv subjects

Zach Hunter

Publications and source records attributed to Zach Hunter.

At least 19 recordsLinked to original sources

arXiv2026

Distinguishability threshold for random geometric graphs

The spherical random geometric graph $G(n,d,p)$ is obtained by sampling $n$ independent points uniformly on the unit sphere $\mathbb{S}^{d-1}\subseteq\mathbb{R}^d$ and joining pairs of points which are sufficiently close, where the threshold is chosen so that the edge probability is $p$. The central question related to this model, and to a broad class of other models, is the following: when does the underlying geometry affect the resulting graph in a way which makes it distinguishable from the Erd\H{o}s--R\'enyi random graph $G(n,p)$, as measured in total variation distance? The precise answer to this question was conjectured by Bubeck, Ding, Eldan, and R\'acz, who predicted that $G(n,d,p)$ and $G(n,p)$ are indistinguishable precisely when $d \gg n^3p^3(\log p^{-1})^3$, and provided a test for distinguishing these models in the low-dimensional regime. Although this conjecture attracted considerable attention from researchers in probability, theoretical computer science, and high-dimensional statistics, it was previously fully proved only in the constant-density case. In this paper, we resolve the distinguishability conjecture in the broad range $1/3 \geq p \geq n^{-1/5} \text{polylog}(n)$. The key ingredient of our proof is a stronger statement which gives a precise asymptotic formula for the probability that $G(n,d,p)$ realizes a prescribed graph $H$: above the conjectured threshold, this probability is at most $(1+o(1))$ times the corresponding probability for $G(n,p)$, with the signed triangle count of $H$ appearing as the leading correction term.

math.PR↗

arXiv2026

Gilbreath's conjecture: a Cram\'er random model and a deterministic analysis

Gilbreath's conjecture asserts that if one starts with the sequence of primes and takes successive absolute differences to create a triangular array, then the left diagonal of this array consists entirely of ones after the first row. In this paper, we show that the analogue of this conjecture for a Cram\'er random model holds, in which the (normalized) prime gaps are replaced by independent random variables with geometric distributions of logarithmic size. We also give some preliminary analysis of the associated continuous probabilistic model for this problem, as well as a deterministic "inverse theorem" that isolates the specific obstructions to Gilbreath's conjecture (assuming a Cram\'er type bound on prime gaps), namely long blocks of zeroes, or very long shallow $\{0,d\}$-valued blocks for some $d \geq 2$.

math.CO↗

arXiv2026

Nearly tight bounds for induced subdivisions

Subdivisions of complete graphs play a central role in combinatorics, having deep connections to structural, extremal, and topological aspects of graph theory. A celebrated conjecture of Mader, proved independently by Bollob\'as and Thomason and by Koml\'os and Szemer\'edi, states that every graph of average degree of order $h^2$ contains a subdivision of $K_h$. In this paper, we consider the induced variant of this problem. A theorem of K\"uhn and Osthus implies that, for every fixed graph $H$ and every $s\ge 1$, graphs of sufficiently large average degree contain either a copy of $K_{s,s}$ or an induced subdivision of $H$. However, even for $H=K_h$, the best previous quantitative bounds were far from optimal. We prove nearly tight bounds for forcing induced subdivisions of $K_h$. We show that every $K_{s,t}$-free graph of average degree $\Omega_{s,t}(h^{2(s-1)}\log^{7(s-1)} h)$ contains an induced subdivision of $K_h$, and that every $C_{2k}$-free graph with $k \geq 3$ and average degree $\Omega_k(h\log^5 h)$ contains an induced subdivision of $K_h$. These bounds substantially improve the previously known results and are nearly optimal in both settings. They also hold if $K_h$ is replaced by any other graph on $h$ vertices.

math.CO↗

arXiv2026

Three-color van der Waerden numbers grow super-exponentially

For $k$ sufficiently large, we show that there is a three-coloring of the first $2^{k (\log^* k)/4}$ positive integers without any monochromatic $k$-term arithmetic progressions. Thus, the three-color van der Waerden number $w(k;3)$ grows faster than any exponential in $k$. We further prove a new lower bound on multicolor van der Waerden numbers which resolves a problem of Erd\H{o}s and Graham on canonical van der Waerden numbers.

math.CO↗

arXiv2026

Communication Complexity of Disjointness under Product Distributions

Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).

cs.CC↗

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↗

arXiv2026

Induced subdivisions of $K_{d+1}$ in graphs of high girth

In this paper, we show that for all $k\geq 10^8$, every graph with minimum degree $k$ and girth at least $10^8$ contains an induced subdivision of a $K_{k+1}$. This answers a problem asked by K\"uhn and Osthus (originally attributed to Shi).

math.CO↗

arXiv2026

Large point-line matchings and small Nikodym sets

For any integer $d \geq 2$ and prime power $q$, we construct unexpectedly large induced matchings in the point-line incidence graph of $\mathbb{F}_{q}^{d}$ by leveraging a new connection with the Furstenberg-S\'ark\"ozy problem from arithmetic combinatorics. In particular, we significantly improve the previously well-known baselines when $q$ is prime, showing that $\mathbb{F}_{q}^{2}$ contains matchings of size $q^{1.233}$ and $\mathbb{F}_{q}^{d}$ contains matchings of size $q^{d-o_{d}(1)}$. These results and their proofs have several applications. First, we also obtain new constructions for finite field Nikodym sets in dimension $d \geq 2$, improving recent results of Tao by polynomial factors. For example, when $q$ is prime, we show the existence of Nikodym sets in $\mathbb{F}_q^d$ of size $q^d - q^{d - o_d(1)}$. Second, we construct a new minimal blocking set in $\mathrm{PG}(2,q)$, solving a longstanding problem in finite geometry. Third, we obtain new constructions for the minimal distance problem (in $\mathbb{R}^{2}$ and also in higher dimensions), improving a recent result of Logunov-Zakharov. We also obtain analogous results for general finite fields with large characteristics. In particular, in one of our constructions we introduce a new special set of points inside the norm hypersurface in $\mathbb{F}_{q}^{d}$, which directly generalizes the classical Hermitian unital and which may be of independent interest for applications.

math.CO↗

arXiv2026

A Hal\'asz-type theorem for permutation anticoncentration

Given a set $A=\{a_1,\ldots,a_n\}$ of real numbers and real coefficients $b_1,\ldots,b_n$, consider the distribution of the sum obtained by pairing the $a_i$'s with the $b_i$'s according to a uniformly random permutation. A recent theorem of Pawlowski shows that as soon as the coefficients are not all equal, this distribution is always spread out at scale $n^{-1}$: no single value can occur with probability larger than $\frac{1}{2\lceil n/2\rceil + 1}$, and this bound is sharp in general. We show that stronger anticoncentration holds when the coefficients have additional diversity. We quantify the structure of the coefficient multiset by a simple statistic depending on its multiplicity profile, and prove that the maximum point mass of the permuted sum decays polynomially faster as this statistic grows. In particular, when the coefficients are all distinct we obtain a bound of $n^{-5/2+o(1)}$, which can be regarded as an analogue of a classical theorem of Erd\H{o}s and Moser.

math.CO↗

arXiv2025

Gaussian random graphs and Ramsey numbers

We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds.

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, Koml\'os and Szemer\'edi. 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

Cyclic subsets of tournaments

Let $G$ be a Dirac graph, and let $S$ be a vertex subset of $G$, chosen uniformly at random. How likely is the induced subgraph $G[S]$ to be Hamiltonian? This question, proposed by Erd\H{o}s and Faudree in 1996, was recently resolved by Dragani\'c, Keevash and M\"uyesser, in the setting of graphs. In this paper, we study a similar question for tournaments -- if $T$ is a tournament of high minimum degree, how likely is it for a random induced subtournament of $T$ to be Hamiltonian? We prove an optimal bound on this probability, and extend the results to the regime where the subset is not sampled uniformly at random, but according to a $p$-biased measure.

math.CO↗

arXiv2025

An improved construction for the triangle removal lemma

We construct $n$-vertex graphs $G$ where $\epsilon n^2$ edges must be deleted to become triangle-free, which contain less than $\epsilon^{(C_{\text{new}}-o(1))\log_2 1/\epsilon}n^3$ triangles for $C_{\text{new}}= \frac{1}{4\log_2(4/3)} \approx 1.6601$. Previously, a bound of the same shape was known, but with $C_{\text{new}}$ replaced by $C_{\text{old}} := C_{\text{new}}/2$. Our construction uses ideas from additive combinatorics, drawing especially from the corners problem, but does not yield new bounds for those problems.

math.CO↗

arXiv2025

$C_4$-free subgraphs of high degree with geometric applications

The Zarankiewicz problem, a cornerstone problem in extremal graph theory, asks for the maximum number of edges in an $n$-vertex graph that does not contain the complete bipartite graph $K_{s,s}$. While the problem remains widely open in the case of general graphs, the past two decades have seen significant progress on this problem for various restricted graph classes -- particularly those arising from geometric settings -- leading to a deeper understanding of their structure. In this paper, we develop a new structural tool for addressing Zarankiewicz-type problems. More specifically, we show that for any positive integer $k$, every graph with average degree $d$ either contains an induced $C_4$-free subgraph with average degree at least $k$, or it contains a $d$-vertex subgraph with $\Omega_k(d^2)$ edges. As an application of this dichotomy, we propose a unified approach to a large number of Zarankiewicz-type problems in geometry, obtaining optimal bounds in each case.

math.CO↗

arXiv2025

Lower bounds for Ramsey numbers of bounded degree hypergraphs

We prove that, for all $k \ge 3,$ and any integers $\Delta, n$ with $n \ge \Delta,$ there exists a $k$-uniform hypergraph on $n$ vertices with maximum degree at most $\Delta$ whose $4$-color Ramsey number is at least $\mathrm{tw}_k(c_k \Delta) \cdot n$, for some constant $c_k > 0$, where $\mathrm{tw}_k$ denotes the tower function. For $k \ge 4,$ this is tight up to the constant $c_k$ and for $k = 3$ it is known to be tight up to a factor of $\log \Delta$ on top of the tower. It extends a well-known result of Graham, R\"{o}dl and Ruci\'{n}ski for graphs and answers a question of Conlon, Fox and Sudakov from 2008.

math.CO↗

arXiv2024

Monochromatic odd cycles in edge-coloured complete graphs

It is easy to see that every $q$-edge-colouring of the complete graph on $2^q+1$ vertices must contain a monochromatic odd cycle. A natural question raised by Erd\H{o}s and Graham in $1973$ asks for the smallest $L(q)$ such that every $q$-edge-colouring of $K_{2^q+1}$ must contain a monochromatic odd cycle of length at most $L(q)$. In here, we show that $L(q)=O\left(\frac{2^q}{q^{1-o(1)}}\right)$ giving the first non-trivial upper bound on $L(q)$.

math.CO↗

arXiv2024

Almost-full transversals in equi-$n$-squares

In 1975, Stein made a wide generalisation of the Ryser-Brualdi-Stein conjecture on transversals in Latin squares, conjecturing that every equi-$n$-square (an $n\times n$ array filled with $n$ symbols where each symbol appears exactly $n$ times) has a transversal of size $n-1$. That is, it should have a collection of $n-1$ entries that share no row, column, or symbol. In 2017, Aharoni, Berger, Kotlar, and Ziv showed that equi-$n$-squares always have a transversal with size at least $2n/3$. In 2019, Pokrovskiy and Sudakov disproved Stein's conjecture by constructing equi-$n$-squares without a transversal of size $n-\frac{\log n}{42}$, but asked whether Stein's conjecture is approximately true. I.e., does an equi-$n$-square always have a transversal with size $(1-o(1))n$? We answer this question in the positive. More specifically, we improve both known bounds, showing that there exist equi-$n$-squares with no transversal of size $n-\Omega(\sqrt{n})$ and that every equi-$n$-square contains $n-n^{1-\Omega(1)}$ disjoint transversals of size $n-n^{1-\Omega(1)}$.

math.CO↗