arXiv subjects

Xinqi Huang

Publications and source records attributed to Xinqi Huang.

11 recordsLinked to original sources

arXiv2026

Homomorphism and VC-dimension thresholds: spectra and separations

Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove $δ_{\mathrm{hom}}(K_{1,s,t}) \ge \max\left\{\frac13,\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, $δ_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge2$. More generally, for every $r\ge3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that $δ_χ,δ_{\mathrm{hom}},δ_{\mathrm{VC}}$, and $δ_{\mathrm B}$ are \emph{pairwise distinct}: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.

math.CO↗

arXiv2026

Exact Homomorphism Thresholds Beyond Cliques

The chromatic threshold, originating in a question of Erdős and Simonovits, asks when a linear minimum-degree condition forces bounded chromatic number in H-free graphs. Motivated by a question of Thomassen, the homomorphism threshold asks for the stronger conclusion that every such graph admits a homomorphism to an H-free graph of bounded order. Since the work of Goddard and Lyle determined the clique case, exact homomorphism thresholds for individual non-complete forbidden graphs have remained unknown. In this paper, we extend the clique case to a larger family of forbidden graphs, determining the homomorphism threshold exactly for every graph in this family.

math.CO↗

arXiv2026

On the spectrum and structure of blowup thresholds

The chromatic threshold of Erdős and Simonovits asks when a minimum-degree condition forces every $H$-free graph to have bounded chromatic number. Thomassen's homomorphism threshold strengthens this by requiring a bounded $H$-free homomorphic image. The recently introduced blowup threshold $δ_{\mathrm B}(H)$ asks for a still more rigid conclusion: when must every sufficiently dense maximal $H$-free graph be an actual blowup of a bounded graph? Thus the blowup threshold measures when quotient-level structure can be upgraded to exact bounded-template structure. We show that, although chromatic and homomorphism thresholds are often hard to separate, the stronger blowup threshold diverges from the chromatic threshold in several fundamental ways. First, we prove that $δ_{\mathrm B}(H)>0$ for every non-bipartite graph $H$. Hence, unlike the chromatic threshold, the blowup threshold never vanishes outside the bipartite world. Second, we prove that $δ_{\mathrm B}$ is not monotone under taking induced subgraphs. This shows that the blowup threshold is sensitive to global features of the forbidden graph and cannot be classified by a direct analogue of the monotonicity-based strategy used for chromatic thresholds. Third, we prove that $δ_{\mathrm B}(H)=\frac{1}{4}$ for a natural family of $3$-chromatic constrained blowups of odd cycles. This gives a new exact blowup-threshold value beyond the chromatic-threshold spectrum.

math.CO↗

arXiv2026

World Reasoning Arena

World models (WMs) are intended to serve as internal simulators of the real world that enable agents to understand, anticipate, and act upon complex environments. Existing WM benchmarks remain narrowly focused on next-state prediction and visual fidelity, overlooking the richer simulation capabilities required for intelligent behavior. To address this gap, we introduce WR-Arena, a comprehensive benchmark for evaluating WMs along three fundamental dimensions of next world simulation: (i) Action Simulation Fidelity, the ability to interpret and follow semantically meaningful, multi-step instructions and generate diverse counterfactual rollouts; (ii) Long-horizon Forecast, the ability to sustain accurate, coherent, and physically plausible simulations across extended interactions; and (iii) Simulative Reasoning and Planning, the ability to support goal-directed reasoning by simulating, comparing, and selecting among alternative futures in both structured and open-ended environments. We build a task taxonomy and curate diverse datasets designed to probe these capabilities, moving beyond single-turn and perceptual evaluations. Through extensive experiments with state-of-the-art WMs, our results expose a substantial gap between current models and human-level hypothetical reasoning, and establish WR-Arena as both a diagnostic tool and a guideline for advancing next-generation world models capable of robust understanding, forecasting, and purposeful action. The code is available at https://github.com/MBZUAI-IFM/WR-Arena.

cs.CV↗

arXiv2026

Colour diversity in spanning structures under Dirac-type conditions

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains a Hamilton cycle with $n - O(n^{1/2})$ distinct colours. In the bipartite setting, the analogous question for perfect matchings is closely related to permutations in Latin squares. In this paper, we investigate how a Dirac-type minimum degree condition forces colour diversity in spanning structures. For every constant $1/2 < c \le 1$, we prove the following. $\bullet$ Every properly edge-coloured graph $G$ on $n$ vertices with $δ(G)\ge cn$ contains a Hamilton cycle with at least $cn - O(n^{1/2})$ distinct colours. $\bullet$ Every subset of an $n\times n$ Latin square with at least $cn$ cells in each row and each column contains a permutation with at least $cn - O(n^{2/3})$ distinct symbols. Both bounds are best possible up to the error term.

math.CO↗

arXiv2025

Quantitative Frameproof Codes and Hypergraphs

Frameproof codes are a class of secure codes introduced by Boneh and Shaw in the context of digital fingerprinting, and have been widely studied from a combinatorial point of view. In this paper, we study a quantitative extension of frameproof codes and hypergraphs, referred to as {\it quantitative frameproof codes and hypergraphs}. We give asymptotically optimal bounds on the maximum sizes of these structures and determine their exact sizes for a broad range of parameters. In particular, we introduce a generalized version of the Erdős matching number in our proof and derive relevant estimates for it.

math.CO↗

arXiv2025

Do Vision-Language Models Have Internal World Models? Towards an Atomic Evaluation

Internal world models (WMs) enable agents to understand the world's state and predict transitions, serving as the basis for advanced deliberative reasoning. Recent large Vision-Language Models (VLMs), such as OpenAI o3, GPT-4o and Gemini, exhibit potential as general-purpose WMs. While the latest studies have evaluated and shown limitations in specific capabilities such as visual understanding, a systematic evaluation of VLMs' fundamental WM abilities remains absent. Drawing on comparative psychology and cognitive science, we propose a two-stage framework that assesses Perception (visual, spatial, temporal, quantitative, and motion) and Prediction (mechanistic simulation, transitive inference, compositional inference) to provide an atomic evaluation of VLMs as WMs. Guided by this framework, we introduce WM-ABench, a large-scale benchmark comprising 23 fine-grained evaluation dimensions across 6 diverse simulated environments with controlled counterfactual simulations. Through 660 experiments on 15 latest commercial and open-source VLMs, we find that these models exhibit striking limitations in basic world modeling abilities. For instance, almost all models perform at near-random accuracy when distinguishing motion trajectories. Additionally, they lack disentangled understanding -- e.g., some models tend to believe blue objects move faster than green ones. More rich results and analyses reveal significant gaps between VLMs and human-level world modeling.

cs.CL↗

arXiv2025

Largest dyadic dual VC-dimension of non-piercing families

The dyadic dual VC-dimension of a set system $ \mathcal{F} $ is the largest integer $ \ell $ such that there exist $ \ell $ sets $ F_1, F_{2}, \dots, F_\ell \in \mathcal{F} $, where every pair $ \{i, j\} \in \binom{[\ell]}{2} $ is witnessed by an element $ a_{i,j} \in F_i \cap F_j $ that does not belong to any other set $ F_k $ with $ k \in [\ell] \setminus \{i, j\} $. In this paper, we determine the largest dyadic dual VC-dimension of a non-piercing family is exactly $4$, providing a rare example where the maximum of this parameter can be determined for a natural family arising from geometry. As an application, we give a short and direct proof that the transversal number $ τ(\mathcal{F}) $ of any non-piercing family is at most $Cν(\mathcal{F})^9 $, where $ ν(\mathcal{F}) $ is the matching number and $C$ is a constant. This improves a recent result of Pálvölgyi and Zólomy.

math.CO↗

arXiv2025

Interpolating chromatic and homomorphism thresholds

The problem of chromatic thresholds seeks for minimum degree conditions that ensure $H$-free graphs to have a bounded chromatic number, or equivalently a bounded size homomorphic image. The strengthened homomorphism thresholds problem further requires that the homomorphic image itself is $H$-free. The purpose of this paper is two-fold. First, we define a generalized notion of threshold which encapsulates and interpolates chromatic and homomorphism thresholds via the theory of VC-dimension. Our first result shows a smooth transition between these two thresholds when varying the restrictions on homomorphic images. In particular, we proved that for $t \ge s \ge 3$ and $ε>0$, if $G$ is an $n$-vertex $K_s$-free graph with VC-dimension $d$ and $δ(G) \ge (\frac{(s-3)(t-s+2)+1}{(s-2)(t-s+2)+1} + ε)n$, then $G$ is homomorphic to a $K_t$-free graph $H$ with $|H| = O(1)$. Moreover, we construct graphs showing that this minimum degree condition is optimal. This extends and unifies the results of Thomassen, Łuczak and Thomassé, and Goddard, Lyle and Nikiforov, and provides a deeper insight into the cause of existences of homomorphic images with various properties. Second, we introduce the blowup threshold $δ_B(H)$ as the infimum $α$ such that every $n$-vertex maximal $H$-free graph $G$ with $δ(G)\geαn$ is a blowup of some $F$ with $|F|=O(1)$. This notion strengthens homomorphism threshold. While the homomorphism thresholds for odd cycles remain unknown, we prove that $δ_B(C_{2k-1})=1/(2k-1)$ for any integer $k\ge 2$. This strengthens the result of Ebsen and Schacht and answers a question of Schacht and shows that, in sharp contrast to the chromatic thresholds, 0 is an accumulation point for blowup thresholds. Our proofs mix tools from VC-dimension theory and an iterative refining process, and draw connection to a problem concerning codes on graphs.

math.CO↗

arXiv2024

Sublinear hitting sets for some geometric graphs

For an $n$-vertex graph $G$, let $h(G)$ denote the smallest size of a subset of $V(G)$ such that it intersects every maximum independent set of $G$. A conjecture posed by Bollobás, Erdős and Tuza in early 90s remains widely open, asserting that for any $n$-vertex graph $G$, if the independence number $α(G) =Ω(n) $, then $h(G) = o(n)$. In this paper, we establish the validity of this conjecture for various classes of graphs, Our main contributions include: \begin{enumerate} \item We provide a novel unified framework to find sub-linear hitting sets for graphs with certain locally sparse properties. Based on this framework, we can find hitting sets of size at most $O(\frac{n}{\log{n}})$ in any $n$-vertex even-hole-free graph (in particular, chordal graph) and in any $n$-vertex disk graph, with linear independence numbers. \item Utilizing geometric observations and combinatorial arguments, we show that any $n$-vertex circle graph $G$ with linear independence number satisfies $h(G)\le O(\sqrt{n})$. Moreover, we extend this methodology to more general classes of graphs. \item We show the conjecture holds for those hereditary graphs having sublinear balanced separators. \end{enumerate} We also show that $h(G)$ can be upper bounded by constants for several sporadic families of graphs with large independence numbers.

math.CO↗

arXiv2024

Focal-free uniform hypergraphs and codes

Motivated by the study of a variant of sunflowers, Alon and Holzman recently introduced focal-free hypergraphs. In this paper, we show that there is an interesting connection between the maximum size of focal-free hypergraphs and the renowned Erdős Matching Conjecture on the maximum number of edges that can be contained in a uniform hypergraph with bounded matching number. As a consequence, we give asymptotically optimal bounds on the maximum sizes of focal-free uniform hypergraphs and codes, thereby significantly improving the previous results of Alon and Holzman. Moreover, by using the existentce results of combinatorial designs and orthogonal arrays, we are able to explicitly determine the exact sizes of maximum focal-free uniform hypergraphs and codes for a wide range of parameters.

math.CO↗