arXiv subjects

Hong Liu

Publications and source records attributed to Hong Liu.

At least 19 recordsLinked to original sources

arXiv2026

Minimum-degree thresholds beyond bounded colourability

We survey recent developments on minimum-degree thresholds in extremal graph theory, with an emphasis on structural refinements of the chromatic-threshold problem. The chromatic threshold asks when a linear minimum-degree condition forces bounded chromatic number in an $H$-free graph; its complete classification shows a striking finite trichotomy. We discuss several stronger and related threshold notions in which bounded colourability is replaced by bounded $H$-free homomorphic images, bounded neighbourhood complexity via VC-dimension, exact blowup structure for maximal $H$-free graphs, and improved quantitative bounds in graph removal lemmas. In contrast with the rigidity of the chromatic threshold, these newer parameters exhibit richer behaviour, including separations from $δ_χ(H)$, new exact values, accumulation points in fixed-chromatic spectra, positivity phenomena for 3-chromatic graphs, asymmetric variants, and failures of monotonicity.

math.CO↗

arXiv2026

Strong counterexamples to Mubayi's supersaturation conjecture in every uniformity

The supersaturation problem asks, for a fixed $r$-graph $\mathcal F$, for the minimum number of copies of $\mathcal F$ in an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges. Mubayi conjectured a local form of supersaturation under a stability hypothesis: if $\mathcal F$ is non-$r$-partite and stable, meaning roughly that the extremal $\mathcal F$-free construction is unique and all near-extremal $\mathcal F$-free $r$-graphs are close to it, then this minimum should be at least $q c(n,\mathcal F)$, where $c(n,\mathcal F)$ is the minimum number of copies created by adding one edge to the extremal $\mathcal F$-free $r$-graph. We disprove this conjectured local lower bound in the strongest possible quantitative sense that the ratio to the conjectured lower bound can be made arbitrarily small. More precisely, for every $r\ge2$ and every $K>1$, we construct a stable non-$r$-partite $r$-graph $\mathcal F$ such that, for all sufficiently large $n$ and every $1\le q\leδn$, there is an $n$-vertex $r$-graph with $\ex(n,\mathcal F)+q$ edges containing at most $K^{-1}q c(n,\mathcal F)$ copies of $\mathcal F$. This arbitrarily large failure already occurs when $q=1$.

math.CO↗

arXiv2026

HARTS: A Prefix-Sharing Training System for Agentic RL with Hybrid-Attention Models

Agentic reinforcement learning (RL) generates irregular rollout trees, but trajectory-by-trajectory training recomputes their shared interaction histories. Existing training-time prefix-reuse systems do not jointly address state recovery, dense execution, and distributed training for hybrid-attention models. We present HARTS (Hybrid-Attention RL over Tree Structures), to our knowledge the first system to demonstrate training acceleration from arbitrary rollout-tree prefix sharing on real hybrid-attention models. A two-level planner constructs compression-oriented microbatches, then jointly refines their membership, cross-microbatch sharing, placement, and execution order. Within each sharing window, gradients from dependent microbatches accumulate before one backward pass through the shared prefix computation. Differentiable boundary-state handoff and bounded replay preserve the per-trajectory chunkwise numerical path, while a linear-time planner attains the global minimum number of sequential linear-attention calls under our packed chunkwise-call model. The runtime uses a common compact layout across attention and dense or mixture-of-experts (MoE) layers, with support for distributed parallelism and activation recomputation. Recovering log-probabilities for each original trajectory position preserves RL objectives; for deterministic top-k routing without token dropping, occurrence counts restore MoE objective weights and expert-load statistics. On SWE-bench, HARTS delivers 4.77-10.45x the update throughput of trajectory-expanded training across two sampling schemes on 7.9B- and 124B-parameter hybrid-attention MoE models using 8-64 GPUs. On the 7.9B model, full-model logit differences are comparable to baseline self-rerun variation. The 124B model shows stable online learning trends through step 150 on SWE-bench and $τ^3$-Bench.

cs.LG↗

arXiv2026

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 $G$ to 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 with $m$ 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 extension 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 with $m$ edges with $λ^2(G) \ge (1- \frac{1}{r} - o(1))2m$, 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 both the classical Erdős-Simonovits stability and the Nikiforov vertex-spectral stability. This edge-spectral stability theorem is the key ingredient in our companion work on the Brualdi-Hoffman-Turán problem for color-critical forbidden graphs. 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. The constant $\frac{1}{2}$ is best possible as witnessed by a random construction.

math.CO↗

arXiv2026

The asymptotic Erd{ő}s--Frankl--Füredi cover-free conjecture

In 1985, Erd{ő}s, Frankl and F{ü}redi conjectured that the threshold $T(r)$ for the existence of an $r$-cover-free family with more members than points satisfies $T(r)=(1+o(1))r^2$. We confirm this conjecture by proving $T(r)\ge r^2+r+2$ for every $r\ge1$.

math.CO↗

arXiv2026

Colored Turán densities and finite directed templates

Classical Turán theory relates the asymptotic extremal problem for an ordinary forbidden graph to its chromatic number. This principle fails when a forbidden copy must respect prescribed vertex colors, as occurs naturally in multipartite extremal problems and in embeddings with prescribed clusters. We study properly vertex-colored graphs whose hosts have a prescribed density between every pair of color classes, with containment interpreted through color-preserving homomorphisms. Our main theorem gives a finite directed reduction: for every fixed $m$ and $\varepsilon>0$, every sufficiently many-colored $d$-dense host admits a colored homomorphism from an $m$-layer half-blowup $D[m]$ of a finite digraph $D$ with density at least $d-\varepsilon$, where $D$ may be chosen with $\lceil\varepsilon^{-1}\rceil$ vertices. Consequently, the colored Turán density of every non-empty finite forbidden family equals the homomorphism Turán density of its associated acyclic orientations. We then determine the complete finite value set: these densities are exactly the Lagrangians of finite loopless digraphs, and hence are rational. The main technical input is a low-resolution sampling method that preserves all pairwise bipartite degree profiles on a large subfamily of color classes while keeping the sample size independent of the number of retained classes.

math.CO↗

arXiv2026

A proof of the rational exponents conjecture

The rational exponents conjecture of Erdős and Simonovits asserts that for every rational $r\in [1,2]$ there exists a graph $H$ such that $\mathrm{ex}(n,H)=Θ(n^r)$. We present a proof of this conjecture claimed by Epoch AI, explaining how it builds upon ideas and methods developed by many researchers.

math.CO↗

arXiv2026

Asymptotics of the Brown--Erdős--Sós problem at integer exponents

The Brown--Erdős--Sós problem is a fundamental problem in sparse hypergraph Turán theory. For integers $r,k\ge 2$ and $s\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In 1971, Brown, Erdős, and Sós proved that $f^{(r)}\bigl(n;(r-t)k+t,k\bigr)=Θ(n^t)$ for all $r>t\ge 2$ and $k\ge 2$. However, the existence and the value of the leading coefficient have remained largely open. We determine the coefficient $π(r,t,k):=\lim_{n\to\infty}n^{-t}f^{(r)}\bigl(n;(r-t)k+t,k\bigr)$ for every such $r,t,k$, except when $(r,t)=(3,2)$ and $k\ge 4$ is even. In particular, \[ π(r,t,k) = \begin{cases} \frac{2}{t!\bigl(2\binom{r}{t}-1\bigr)}, & \text{if $k$ is odd},\\ \frac{1}{t!\binom{r}{t}}, & \text{if $k$ is even and $r\ge4$}. \end{cases} \] Surprisingly, for $r\ge4$, the limit $π(r,t,k)$ depends on $k$ only through its parity, not its value. For the remaining case, we prove that $π(3,2,k)>1/6$ for all even $k\ge 4$, showing that the natural packing construction is never asymptotically optimal. As an application, we extend a connection of Bennett, Cushman, and Dudek to arbitrary uniformity to resolve several cases of the Erdős--Gyárfás--Shelah generalized Ramsey problem.

math.CO↗

arXiv2026

De-biasing Skeleton-based Action Recognition with Convex Hull Adaptive Shift

Skeleton sequences can represent both individual actions and multi-entity interactions, encompassing human bodies, hands, objects, and robots. Existing approaches to recognize skeleton-based actions and interactions usually adopt a late fusion strategy, which expects individuals are independent and identically distributed to train a robust weight-shared entity encoder. However, observed entity bias in various skeletal data violates this assumption, leading to suboptimal optimization of backbone models that might produce wrong recognition results. This bias arises from the world coordinate system's initial configuration, where the choice of origin often creates bias in representation. To this end, we propose a Convex Hull Adaptive Shift based normalization method to reduce Entity bias (CHASE), improving performance across a variety of skeleton-based action and interaction recognition tasks. To adaptively apply plausible shifts to the input skeletons, we formulate a plug-and-play parameterized network that ensures the relocated world origin lies within the skeleton convex hull, which avoids non-convergence by limiting the search space. To further minimize entity bias, we incorporate an auxiliary objective that leverages pair-wise distribution distances to guide network optimization. To support both single- and multi-entity actions, we propose a sub-entity strategy that offers a consistent formulation for both scenarios. Moreover, CHASE demonstrates compatibility with various intra-skeleton modalities, such as bones and velocities, highlighting its adaptability. Essentially, our method works as a normalization approach to reduce entity bias, enabling subsequent classifiers to achieve improved recognition performance across diverse settings. Extensive experiments on 7 datasets verify our approach by seamlessly integrating with various backbones and significantly boosting their performance.

cs.CV↗

arXiv2026

"Filtering" CFTs at large N: Euclidean Wormholes, Closed Universes, and Black Hole Interiors

Despite its successes, the large-$N$ holographic dictionary remains incomplete. Key features of gravitational path integrals--most notably Euclidean wormholes and the associated failure of factorization--lack a clear interpretation in the standard large-$N$ framework. A related challenge is the possibility of erratic $N$-dependence in CFT observables, behavior with no evident semiclassical gravitational counterpart. We argue that these puzzles point to a missing ingredient in the dictionary: a large-$N$ filter. This filter projects out the erratic $N$-dependence of CFT quantities when mapping them to semiclassical bulk physics, providing an intrinsic boundary definition of gravitational "averages." It also offers a boundary explanation of wormhole contributions and a boundary prediction of their amplitudes, thereby giving a natural resolution of the factorization puzzle. In addition, we derive an infinite tower of inequalities constraining wormhole amplitudes and argue that internal wormholes do not induce random couplings in the low-energy effective theory. Beyond resolving factorization, the large-$N$ filter supplies a generalized framework from which richer Lorentzian spacetime structures can emerge, including closed universes and black hole interiors. We argue that, as a consequence of erratic large-$N$ behavior, both closed universes and black hole interiors are quantum volatile, and that an AdS spacetime entangled with a baby universe is likewise quantum volatile. This volatility may allow an observer in AdS to infer the existence of the baby universe, whereas for an infalling observer, the ability to make measurements near a black hole horizon may become fundamentally limited--even if they may not live long enough to notice.

hep-th↗

arXiv2026

Ramp, Plateau, and Wormholes without Averaging, and Hyper-non-perturbative Structures in Gravity

Universal hallmarks of quantum chaos---such as the ramp and plateau in the spectral form factor---and the ramp's gravitational duals involving wormholes are widely interpreted as consequences of spectral or ensemble averaging. In this paper, following an earlier proposal of~\cite{Liu25c}, we develop an alternative approach: these phenomena arise as macroscopic smooth structures hidden within erratic microscopic data, which can be isolated through a smooth filter projection. Using the semiclassical Gutzwiller trace formula as a paradigmatic example, we illustrate how many features characteristic of random matrix models---including the ramp, the plateau, the spectral curve, and single-eigenvalue instantons---can be derived in the semiclassical limit without invoking ensemble or explicit spectral averages. We postulate the existence of a minimal Gutzwiller-like structure in the large-$N$ limit of holographic systems and explore its consequences. Beyond deriving the ramp and the plateau, this Gutzwiller-like structure predicts universal rapid macroscopic oscillations in the density of states and the possible existence of hyper-instantons, both of which involve double exponentials in $1/N^2$. On the gravity side, we demonstrate how spacetime wormholes enable the construction of emergent hyper-non-perturbative objects---such as baby-universe and wormhole condensates---which yield double exponential effects in $G_N$. This mirrors the postulated boundary Gutzwiller-like structure and provides a dual gravitational derivation of the universal rapid macroscopic oscillations in the density of states and the spectral plateau.

hep-th↗

arXiv2026

Solution of uniform Turán's Tetrahedron Problem

Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erdős. In the 1980s, Erdős and Sós asked to determine Turán densities of $K_4^{(3)-}$ (broken tetrahedron) and $K_4^{(3)}$ (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Turán density of $K_4^{(3)}$ is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.

math.CO↗

arXiv2026

TEMA: Evidence-Grounded Temporal Question Answering in Multi-Turn Multi-Audio Dialogs

Multi-turn, multi-audio temporal question answering requires models to track target events across follow-up questions, recording switches, and historical references, recovering complete instances and their boundaries for temporal calculation and comparison. We propose TEMA, which connects event perception with evidence-based answering through Route, specifying the audio scope, and Span, describing all relevant intervals as conditional audio captions. We construct TEMA-Dialog with 40,704 dialogs and per-turn evidence and answer supervision, and TEMA-Bench for joint evaluation of evidence and final answers. Training combines temporal grounding initialization, full-dialog supervised fine-tuning, and completeness-first Span-only GRPO. Experiments on Qwen2.5-Omni and AF-Next show improved temporal question answering, particularly event localization and cross-audio comparison. Reinforcement learning applied solely to evidence further improves interval recovery and answer accuracy.

cs.SD↗

arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗

arXiv2026

Small circumference in regular sublinear expanders

Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] proved the existence of nearly Hamilton cycles in sufficiently dense regular sublinear expanders. Montgomery [ICM 2026] subsequently conjectured that, every $d$-regular sublinear expander with $d$ sufficiently large (but constant) is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with $d=\left(\frac12+o(1)\right)\log^2 n$, which can forbid any cycle covering an arbitrarily small given positive constant portion of its vertices. The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks and keeps the other side as a sparse vertex separator. We also prove a similar statement for a closely related notion of edge expanders. For every sufficiently small $γ>0$, there is an infinite family of $n$-vertex $d$-regular $γ$-edge-expanders with $d=Θ(γ^{-1})$ and circumference $O(γn)$, matching the standard lower bound $Ω(γn)$. The construction also comes from an expanding regular core such that each vertex has an almost-complete graph attached to it. The core guarantees edge expansion, while the single-vertex attachments confine every cycle.

math.CO↗

arXiv2026

HunyuanOCR-1.5: Making Lightweight OCR VLMs Faster and Better

We present HunyuanOCR-1.5, a lightweight end-to-end OCR-specialized vision-language model. HunyuanOCR unifies document parsing, text spotting, information extraction, text-image translation, and multi-image document understanding within a single end-to-end VLM. Building upon the lightweight architecture of HunyuanOCR-1.0, HunyuanOCR-1.5 does not redesign the backbone, but systematically improves both efficiency and capability. For efficiency, we adapt DFlash to OCR decoding, significantly reducing the latency of long structured outputs such as dense documents, tables, and formulas while preserving output distribution. Powered by DFlash, HunyuanOCR-1.5 achieves a 6.37x Transformer inference speedup and a 2.14x speedup under vLLM, delivering the fastest inference among lightweight OCR VLMs. For capability, we propose Agentic Data Flow, an agent-driven data construction system that transforms model weaknesses into executable data requirements and autonomously performs material search, quality verification, and pipeline development. It substantially improves long-tail capabilities in ancient-script OCR, fine-grained chart and table parsing, multi-image text-centric QA, low-resource multilingual parsing, and document hallucination evaluation. HunyuanOCR-1.5 ranks among the top-tier end-to-end OCR solutions on OmniDocBench v1.6 while achieving new performance milestones across these long-tail tasks. Combined with an upgraded pretraining and post-training recipe, HunyuanOCR-1.5 further extends its capability in high-resolution, long-context, and multi-task scenarios. Experiments demonstrate faster inference, broader OCR capability coverage, and the deployment advantages of a lightweight end-to-end model. We will release the model weights and training code to support future research and real-world OCR applications.

cs.CV↗

arXiv2026

D-Quant: Driftable Entropy Coding for KV Cache Quantization

The KV cache has become a major bottleneck in deploying LLMs, as its memory footprint grows linearly with sequence length and batch size, imposing substantial pressure on both memory capacity and bandwidth. Among various KV cache compression techniques, quantization is particularly attractive due to its effectiveness and ease of deployment. However, most existing methods rely on fixed-width quantization, where a $b$ bit representation is inherently limited to $2^b$ quantization levels. As the bit width decreases, the number of available levels shrinks exponentially, leading to severe information loss and rapid performance degradation. We further observe that fixed-width quantization fails to exploit the highly non-uniform distribution of KV cache. After rotation and normalization, KV values approximately follow a normal distribution, with most values concentrated near the center and only a small fraction appearing in the tails. Nevertheless, fixed-width coding allocates the same number of bits to frequent and rare symbols. Entropy coding naturally exploits such non-uniformity by assigning shorter codewords to frequent symbols and longer ones to rare symbols, substantially reducing the average number of bits required for representation. However, its variable-length output is not suited to highly parallel attention kernels, where efficient dequantization and computation rely on regular memory layouts and fixed-stride accesses. To bridge this gap, we propose \textbf{D-Quant}, a flexible KV cache quantization framework that introduces a \textbf{drift} mechanism to convert entropy-coded representations of each token into fixed-size bitstreams, enabling regular memory access and parallel dequantization within attention kernels.

cs.CL↗

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↗