Searcharxiv⌕ Search

arXiv subjects

Shengtong Zhang

Publications and source records attributed to Shengtong Zhang.

At least 19 recordsLinked to original sources
arXiv2026

Edge-spectral Turán theorems for color-critical graphs with applications

A classical result of Nosal asserts that every $m$-edge graph with spectral radius $λ(G)> \sqrt{m}$ contains a triangle. A celebrated extension of Nikiforov [2002, CPC] states that if $G$ is an $m$-edge graph with $λ(G)> \sqrt{(1- {1}/{r})2m}$, then $G$ contains a clique $K_{r+1}$. This result implies the Turán theorem and Wilf theorem, and offers a new perspective on the existence of substructures. The edge-spectral conditions are versatile for enforcing substructures, as they can be applied to any sparse graph regardless of its edge density. In this paper, we prove that for any color-critical graph $F$ with chromatic number $χ(F)=r+1\ge 4$, if $m$ is sufficiently large and $G$ is an $F$-free graph with $m$ edges, then $λ(G)\le \sqrt{(1- {1}/{r})2m}$, with equality if and only if $G$ is a regular complete $r$-partite graph. This settles an open problem proposed by Yu and Li [2025, arXiv] and also gives spectral bounds for graphs forbidding books and wheels. Secondly, we establish an asymptotic formula and structural characterization when we forbid an almost-bipartite graph $F$, where $F$ is called almost-bipartite if it can be made bipartite by removing at most one edge. As applications, we determine the unique $m$-edge spectral extremal graph for every integer $m$ when avoiding certain substructures, including complete bipartite graphs plus an edge, cycles plus an edge, and theta graphs, etc. Our results resolve an open problem proposed by Li, Zhao and Zou [2025, JGT], as well as two conjectures posed by Liu and Li [2025, LAA]. The arguments in our proofs are based on the edge-spectral stability method recently established by the authors. In addition, we develop some new spectral techniques, including the stability result for the Perron--Frobenius eigenvector.

math.CO↗
arXiv2026

Maximising the number of solutions to linear equations

We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $γ_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $γ_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le γ_{1,1,-3}\le 1/2-δ$ for some constant $δ>0$. In addition we show that the limit as $a \rightarrow \infty$ of $γ_{1,1,-a}$ is equal to precisely $1/5$.

math.NT↗
arXiv2026

The Problem Is the Problem: Towards Scalable Mathematical Discovery

AI systems are increasingly capable of contributing to mathematical research. In research practice, frontier-model reasoning is a limited resource, and expert mathematical review is even more sharply constrained. Allocating these scarce resources well is therefore central to making AI-assisted mathematical discovery efficient. In most current AI-for-math workflows, human effort is concentrated at the beginning and end, in selecting suitable research problems and later reviewing the resulting artifacts. These two stages are becoming bottlenecks for research-level mathematics. We address them by proposing a new human-AI discovery paradigm. The human input is no longer a single problem selected in advance, but a research direction in which the experts have interest and expertise. The system then searches a broad literature corpus for candidate problems in that direction. Inspired by search and recommender systems, we build Find, Attempt, and Recommend (FAR), a literature-to-review cascade that automates the search for suitable problems and focuses human attention on artifacts that have passed several stages of filtering. In a combinatorics pilot, the pipeline starts from 5,245 combinatorics papers, recovers 6,453 candidate conjectures or open problems, and filters them to 4,717 apparently well-posed and still-open conjectures. Subsequent reasoning and automated triage stages surface 598 potential resolutions and select 77 items for author-team review. Among them, we identify many interesting discoveries, including results on conjectures and questions of Davies--Jenssen--Perkins--Roberts, Erdős--Straus, Ikenmeyer--Pak--Panova, and Lund--Saraf--Wolf. These results demonstrate the effectiveness of this new mode of human-AI collaboration for mathematical discovery.

cs.AI↗
arXiv2026

The positive and negative square-energy conjecture

Let $s^+(G)$ and $s^-(G)$ denote the sums of the squares of the positive and negative adjacency eigenvalues of a graph $G$, respectively. We prove the conjecture of Elphick, Farber, Goldberg, and Wocjan that every connected graph $G$ on $n$ vertices satisfies $$ \min\{s^+(G),s^-(G)\}\ge n-1. $$ The proof introduces a new framework for square-energy estimates, in which the Hadamard squares of positive semidefinite matrices that encode these spectral quantities are relaxed to the full doubly nonnegative cone.

math.CO↗
arXiv2026

A positive square-energy strengthening of Turán's theorem

Let $G$ be an $n$-vertex graph with clique number $ω(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{ω(G)}\right)n. $$ This strengthens Wilf's classical spectral Turán theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin--Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro--Wei greedy algorithm on the complement.

math.CO↗
arXiv2026

Irrationality of rapidly converging series: a problem of Erdős and Graham

Answering a question of Erdős and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/ϕ^n}=\infty$ for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $ϕ$ denotes the golden ratio. We also provide a positive generalization to $\sum_{n=1}^\infty 1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent \emph{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions.

math.NT↗
arXiv2026

Chromatic thresholds for linear equations and recurrence

Motivated by classical problems in extremal graph theory, we study a chromatic analogue of Roth-type questions for linear equations over $\mathbb F_p$. Given a homogeneous equation $\mathcal L:\sum_{i=1}^k c_i x_i=0$ with $k\ge 3$, we study $\mathcal L$-solution-free sets $A\subseteq \mathbb F_p$ through the chromatic number of the Cayley graph $\mathsf{Cay}(\mathbb F_p,A)$. We introduce the \emph{chromatic threshold} $δ_χ(\mathcal L)$, the minimum density that guarantees bounded chromatic number of $\mathsf{Cay}(\mathbb F_p,A)$ among all $\mathcal L$-solution-free sets $A$, and determine exactly when $δ_χ(\mathcal L)=0$. We prove that $δ_χ(\mathcal L)=0$ if and only if $\mathcal L$ contains a zero-sum subcollection of at least three coefficients. A key ingredient is a quantitative chromatic lower bound for Cayley graphs on $\mathbb Z_p^n$ generated by Hamming balls around the all-ones vector. This is obtained by introducing a new Kneser-type graph that admits a natural embedding into $\mathbb Z_p^n$, together with an equivariant Borsuk--Ulam type argument. As a consequence, we resolve a question of Griesmer. We further relate our classification to the hierarchy of measurable, topological, and Bohr recurrence. In particular, we show that every infinite discrete abelian group admits a set that is topological recurrent but not measurable recurrent, extending the seminal examples of Kříž and Ruzsa.

math.CO↗
arXiv2026

The Frankl-Pach upper bound is not tight for any uniformity

For any positive integers $n\ge d+1\ge 3$, what is the maximum size of a $(d+1)$-uniform set system in $[n]$ with VC-dimension at most $d$? In 1984, Frankl and Pach initiated the study of this fundamental problem and provided an upper bound $\binom{n}{d}$ via an elegant algebraic proof. Surprisingly, in 2007, Mubayi and Zhao showed that when $n$ is sufficiently large and $d$ is a prime power, the Frankl-Pach upper bound is not tight. They also remarked that their method requires $d$ to be a prime power, and asked for new ideas to improve the Frankl-Pach upper bound without extra assumptions on $n$ and $d$. In this paper, we provide an improvement for any $d\ge 2$ and $n\ge 2d+2$, which demonstrates that the long-standing Frankl-Pach upper bound $\binom{n}{d}$ is not tight for any uniformity. Our proof combines a simple yet powerful polynomial method and structural analysis.

math.CO↗
arXiv2026

Spectral Sidorenko inequalities and edge-spectral supersaturation

We develop a spectral approach to Sidorenko-type inequalities and apply it to establish sharp edge-spectral supersaturation results. Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: $$ \hom(H,G)\ge M(G)^e |V(G)|^{v-2e} \quad \text{ if and only if }\quad \hom(H,G)\ge λ(G)^{2e-v}M(G)^{v-e}.$$ We also introduce an operator-norm certificate which, via the Riesz--Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. The converse direction in the equivalence theorem is proved by a tensor-power spectral regularization lemma. Our main result provides a unified framework to prove sharp asymptotic edge-spectral supersaturation results for degenerate bipartite graphs with the Sidorenko property, including complete bipartite graphs and even cycles. Let $S_{t-1,m}$ be the split graph with $m$ edges obtained by joining a clique $K_{t-1}$ to an independent set. For every $m$-edge graph $G$ with $λ(G)>λ(S_{t-1,m})$, $$\texttt{#} K_{t,t}(G) \ge \Big(\frac{2^{-(t-1)^2}}{(t!)^2}-o(1)\Big)m^t \quad \text{and}\quad \texttt{#}C_{2t}(G) \ge \Big(\frac{(t-1)!}{2t^t}-o(1)\Big)m^t.$$ Both constants are best possible: the first is attained asymptotically by random graphs, while the second is attained by split graphs. The supersaturation proofs combine spectral Sidorenko inequalities with a heavy-edge pruning process, a Perron-vector localized/delocalized dichotomy, and incidence-matrix inequalities.

math.CO↗
arXiv2026

Bounds on median eigenvalues of graphs of bounded degree

We prove that for every integer $d \ge 3$, the median eigenvalues of any graph of maximum degree $d$ are bounded above by $\sqrt{d-1}$. We also prove that, in three separate cases, the median eigenvalues of a graph of maximum degree $d$ are bounded below by $-\sqrt{d-1}$: when the graph is triangle-free, when $d-1$ is a perfect square, or when $d \ge 75$. These results resolve, for all but finitely many values of $d$, an open problem of Mohar on median eigenvalues of graphs of maximum degree $d$. As a byproduct, we establish an upper bound on the average energy of graphs of maximum degree at most $d$, generalizing a previous result of van Dam, Haemers, and Koolen for $d$-regular graphs.

math.CO↗
arXiv2026

Composer 2 Technical Report

Composer 2 is a specialized model designed for agentic software engineering. The model demonstrates strong long-term planning and coding intelligence while maintaining the ability to efficiently solve problems for interactive use. The model is trained in two phases: first, continued pretraining to improve the model's knowledge and latent coding ability, followed by large-scale reinforcement learning to improve end-to-end coding performance through stronger reasoning, accurate multi-step execution, and coherence on long-horizon realistic coding problems. We develop infrastructure to support training in the same Cursor harness that is used by the deployed model, with equivalent tools and structure, and use environments that match real problems closely. To measure the ability of the model on increasingly difficult tasks, we introduce a benchmark derived from real software engineering problems in large codebases including our own. Composer 2 is a frontier-level coding model and demonstrates a process for training strong domain-specialized models. On our CursorBench evaluations the model achieves a major improvement in accuracy compared to previous Composer models (61.3). On public benchmarks the model scores 61.7 on Terminal-Bench and 73.7 on SWE-bench Multilingual in our harness, comparable to state-of-the-art systems.

cs.SE↗
arXiv2026

Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems

We present a case study in semi-autonomous mathematics discovery, using Gemini to systematically evaluate 700 conjectures labeled 'Open' in Bloom's Erdős Problems database. We employ a hybrid methodology: AI-driven natural language verification to narrow the search space, followed by human expert evaluation to gauge correctness and novelty. We address 13 problems that were marked 'Open' in the database: 5 through seemingly novel autonomous solutions, and 8 through identification of previous solutions in the existing literature. Our findings suggest that the 'Open' status of the problems was through obscurity rather than difficulty. We also identify and discuss issues arising in applying AI to math conjectures at scale, highlighting the difficulty of literature identification and the risk of ''subconscious plagiarism'' by AI. We reflect on the takeaways from AI-assisted efforts on the Erdős Problems.

cs.AI↗
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-Sárközy 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

Coloring small locally sparse degenerate graphs and related problems

The classic upper bound on the chromatic number of $d$-degenerate graphs is $d+1$, shown to be tight by complete graphs. A natural question is whether this bound remains tight if one forbids large cliques. Classic constructions of Tutte and Zykov from the early 50s show that there exist $d$-degenerate $(d+1)$-chromatic graphs that are triangle-free, however these constructions grow rapidly with $d$. Motivated by this and addressing a problem posed by the second author at the Oberwolfach Graph Theory workshop, we prove that the minimum order $f(d)$ of a $d$-degenerate triangle-free graph of chromatic number $d+1$ satisfies $e^{Ω(d)}\le f(d)\le e^{O(d^2\log d)}.$ The lower bound follows from a novel upper bound on the chromatic number of triangle-free graphs: Every triangle-free $d$-degenerate graph $G$ on $n \le e^{O(d)}$ vertices satisfies $$χ(G)\le O\left(\frac{d}{\log\left(d/\log n\right)}\right).$$ We extend this to a more general result about degenerate graphs with sparse neighborhoods, which has applications to many graph coloring problems: For example, we prove that every counterexample to Hadwiger's conjecture with parameter $t$ must have a complete bipartite subgraph with one exponentially large side ($K_{a,b}$ where $a=(\log t)^{1/2-o(1)}$ and $b=e^{t^{1-o(1)}}$) or a small and very dense subgraph (of order $\le t$ with $t^{2-o(1)}$ edges) in some neighborhood. For the upper bound on $f(d)$ we establish a surprising connection between $f(d)$ and the on-line-chromatic number $g(n)$ of $n$-vertex triangle-free graphs. We also give an asymptotic improvement of the previous best upper bound for $g(n)$ due to Lovász, Saks and Trotter from 1989. Along the way we disprove a generalization of Harris' fractional coloring conjecture to graphs of bounded clique number and raise numerous problems which open up interesting directions to explore for future research.

math.CO↗
arXiv2025

Harmonic LCM patterns and sunflower-free capacity

Fix an integer $k\ge 3$. Call a set $A\subseteq [N]$ LCM-$k$-free if it does not contain distinct $a_1,\dots,a_k$ such that $\mathrm{lcm}(a_i,a_j)$ is the same for all $1\le i<j\le k$. Define $$ f_k(N):=\max\left\{\sum_{a\in A}\frac1a: A\subseteq [N] \text{ is LCM-$k$-free}\right\}. $$ Addressing a problem of Erdős, we prove an explicit unconditional lower bound $$ f_k(N)\ge (\log N)^{c_k-o(1)}, \qquad c_k:=\frac{k-2}{e((k-2)!)^{1/(k-2)}}. $$ Let $F_k(n)$ denote the maximum size of a $k$-sunflower-free family of subsets of $[n]$, and define the Erdős--Szemerédi $k$-sunflower-free capacity by $μ_k^{\mathrm S}:=\limsup_{n\to\infty}F_k(n)^{1/n}$. Motivated by a remark of Erdős relating this problem to the sunflower conjecture, we show that $$ (\log N)^{\logμ_k^{\mathrm S}-o(1)} \le f_k(N) \ll (\log N)^{μ_k^{\mathrm S}-1+o(1)}. $$ Furthermore, we show that the Erdős--Szemerédi sunflower conjecture fails for this fixed $k$ (i.e. $μ_k^{\mathrm S}=2$) if and only if $f_k(N)=(\log N)^{1-o(1)}$.

math.NT↗
arXiv2025

From small eigenvalues to large cuts, and Chowla's cosine problem

We prove that every graph with average degree $d$ and smallest adjacency eigenvalue $|λ_n|\leq d^γ$ contains a clique of size $d^{1-O(γ)}$. A simple corollary of this yields the first polynomial bound for Chowla's cosine problem (1965): for every finite set $A\subseteq \mathbb{Z}_{>0}$, the minimum of the cosine polynomial satisfies $$\min_{x\in [0, 2π]}\sum_{a\in A}\cos(ax)\leq -|A|^{1/10-o(1)}.$$ Another application makes significant progress on the problem of MaxCut in $H$-free graphs initiated by Erdős and Lovász in the 1970's. We show that every $m$-edge graph with no clique of size $m^{1/2-δ}$ has a cut of size at least $m/2+m^{1/2+\varepsilon}$ for some $\varepsilon=\varepsilon(δ)>0$.

math.CO↗
arXiv2025

Triangle Ramsey numbers of complete graphs

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total.

math.CO↗
arXiv2025

An edge-spectral Erdős-Stone-Simonovits theorem and its stability

We study the extremal problem that relates the spectral radius $λ(G)$ of an $F$-free graph $G$ with its number of edges. Firstly, we prove that for any graph $F$ with chromatic number $χ(F)=r+1\ge 3$, if $G$ is an $F$-free graph on $m$ edges, then $λ^2(G)\le {(1-\frac{1}{r} + o(1))2m}$. This provides a unified extension of both the Erdős--Stone--Simonovits theorem and its vertex-spectral version due to Nikiforov, and confirms a conjecture proposed by Li, Liu and Feng. We also establish the corresponding edge-spectral stability, showing that if $G$ is an $F$-free graph on $m$ edges with $λ^2(G)=(1- \frac{1}{r} - o(1))2m$, then $G$ differs from a complete bipartite graph by $o(m)$ edges when $r=2$, and $G$ differs from an $r$-partite Turán graph by $o(m)$ edges when $r\ge 3$. This extends the classical Erdős--Simonovits stability theorem. As an application of our method, we improve a result of Zhai, Lin and Shu by showing that if $λ(G)>\sqrt{m}$, then there exist two vertices in $G$ that have at least $\frac{1}{2}\sqrt{m} - O(1)$ common neighbors. This bound is the best possible as witnessed by a random construction.

math.CO↗