arXiv subjects

Zhuo Wu

Publications and source records attributed to Zhuo Wu.

At least 19 recordsLinked to original sources

arXiv2026

Solution of uniform Tur\'an's Tetrahedron Problem

Tur\'an's Tetrahedron Problem asks to determine the Tur\'an density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Tur\'an in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erd\H{o}s. In the 1980s, Erd\H{o}s and S\'os asked to determine Tur\'an 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, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Tur\'an density of $K_4^{(3)}$ is equal to 1/2; this confirms that R\"odl's lower bound construction from 1986 is optimal.

math.CO↗

arXiv2026

Abundance of Unique Subhypergraphs

Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $k=2$, Erd\H{o}s asked whether there exists a constant $\delta>0$ such that $f_2(n)>\delta$ for all $n$, offering \$100 for a proof and \$25 for a disproof. Recently, Brada\v{c} and Christoph answered this question in the negative,, proving that $f_2(n)$ tends to $0$, or equivalently that no $n$-vertex graph contains a positive proportion of all $n$-vertex graphs as unique subgraphs. In this paper we show that the situation is fundamentally different for $k$-uniform hypergraphs with $k\ge3$. In particular, for every fixed integer $k\ge 3$, we prove that $\liminf_{n\to\infty} f_k(n) \ge 2/9$.

math.CO↗

arXiv2026

Chromatic thresholds for pairs of graphs

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density above which every graph that admits a red-blue edge-coloring with no red copy of $H_1$ and no blue copy of $H_2$ has bounded chromatic number. We give a complete answer when both $H_1$ and $H_2$ are 3-chromatic. The threshold takes exactly one of the five values \[ \frac23,\quad \frac57,\quad \frac34,\quad \frac79,\quad \frac45, \] and we characterize precisely which pairs $(H_1,H_2)$ give each value. The classification is determined by the ordinary chromatic thresholds of $H_1$ and $H_2$ and by their embeddability into a hierarchy of $C_5$-type Ramsey configurations.

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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} $\delta_\chi(\mathcal L)$, 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 $\delta_\chi(\mathcal L)=0$. We prove that $\delta_\chi(\mathcal L)=0$ 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 K\v{r}\'i\v{z} and Ruzsa.

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arXiv2025

Edge density and minimum degree thresholds for $H$-free graphs with unbounded chromatic number

The chromatic threshold $δ_χ(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of Allen, Böttcher, Griffiths, Kohayakawa, and Morris determined $δ_χ(H)$ for every graph $H$; in particular, if $χ(H)=r\ge 3$, then $δ_χ(H) \in\{\frac{r-3}{r-2},~\frac{2 r-5}{2 r-3},~\frac{r-2}{r-1}\}$. In this paper we investigate the trade-off between minimum degree and edge density in the critical window around the chromatic threshold. For a fixed graph $H$ with $χ(H)=r$, allowing a constant deficit below $δ_χ(H)$, we prove sharp (up to lower-order terms) upper bounds on the edge density of $n$-vertex $H$-free graphs whose chromatic number diverges. Equivalently, within this degree regime we show that a suitable global bound on the number of edges forces the chromatic number to remain bounded. Our results thus quantify how global edge density can compensate for a deficit in the local minimum-degree condition near $δ_χ(H)$; more specifically, we obtain explicit bounds in two of the three possible cases arising in the trichotomy of $δ_χ(H)$. Our extremal constructions -- based on Erdős graphs and blowups of Borsuk--Hajnal graphs -- show that these bounds are best possible up to $o(n^2)$ terms.

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arXiv2025

Optimizing Knowledge Utilization for Multi-Intent Comment Generation with Large Language Models

Code comment generation aims to produce a generic overview of a code snippet, helping developers understand and maintain code. However, generic summaries alone are insufficient to meet the diverse needs of practitioners; for example, developers expect the implementation insights to be presented in an untangled manner, while users seek clear usage instructions. This highlights the necessity of multi-intent comment generation. With the widespread adoption of Large Language Models (LLMs) for code-related tasks, these models have been leveraged to tackle the challenge of multi-intent comment generation. Despite their successes, state-of-the-art LLM-based approaches often struggle to construct correct relationships among intents, code, and comments within a smaller number of demonstration examples. To mitigate this issue, we propose a framework named KUMIC for multi-intent comment generation. Built upon in-context learning, KUMIC leverages Chain-of-Thought (CoT) to optimize knowledge utilization for LLMs to generate intent-specific comments. Specifically, KUMIC first designs a retrieval mechanism to obtain similar demonstration examples, which exhibit high code-comment consistency. Then, KUMIC leverages CoT to guide LLMs to focus on statements facilitating the derivation of code comments aligned with specific intents. In this context, KUMIC constructs a mapping knowledge chain, linking code to intent-specific statements to comments, which enables LLMs to follow similar reasoning steps when generating the desired comments. We conduct extensive experiments to evaluate KUMIC, and the results demonstrate that KUMIC outperforms state-of-the-art baselines by 14.49\%, 22.41\%, 20.72\%, and 12.94\% in terms of BLEU, METEOR, ROUGE-L, and SBERT, respectively.

cs.SE↗

arXiv2025

Stability with minuscule structure for chromatic thresholds

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $d n$ is bounded by a constant depending only on $H$ and $d$. Allen, B{\"o}ttcher, Griffiths, Kohayakawa, and Morris determined the chromatic threshold for every $H$; in particular, they showed that if $\chi(H)=r\ge 3$, then $\delta_\chi(H) \in\{\frac{r-3}{r-2},~\frac{2 r-5}{2 r-3},~\frac{r-2}{r-1}\}$. While the chromatic thresholds have been completely determined, rather surprisingly the structural behaviors of extremal graphs near the threshold remain unexplored. In this paper, we establish the stability theorems for chromatic threshold problems. We prove that every $n$-vertex $H$-free graph $G$ with $\delta(G)\ge (\delta_\chi(H)-o(1))n$ and $\chi(G)=\omega(1)$ must be structurally close to one of the extremal configurations. Furthermore, we give a stronger stability result when $H$ is a clique, showing that $G$ admits a partition into independent sets and a small subgraph on sublinear number of vertices. We show that this small subgraph has fractional chromatic number $2+o(1)$ and is homomorphic to a Kneser graph defined by subsets of a logarithmic size set; both these two bounds are best possible. This is the first stability result that captures the lower-order structural features of extremal graphs. We also study two variations of chromatic thresholds. Replacing chromatic number by its fractional counterpart, we determine the fractional chromatic thresholds for all graphs. Another variation is the bounded-VC chromatic thresholds, which was introduced by Liu, Shangguan, Skokan, and Xu very recently. Extending work of {\L}uczak and Thomass{\'e} on the triangle case, we determine the bounded-VC chromatic thresholds for all cliques.

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arXiv2024

RecSys Arena: Pair-wise Recommender System Evaluation with Large Language Models

Evaluating the quality of recommender systems is critical for algorithm design and optimization. Most evaluation methods are computed based on offline metrics for quick algorithm evolution, since online experiments are usually risky and time-consuming. However, offline evaluation usually cannot fully reflect users' preference for the outcome of different recommendation algorithms, and the results may not be consistent with online A/B test. Moreover, many offline metrics such as AUC do not offer sufficient information for comparing the subtle differences between two competitive recommender systems in different aspects, which may lead to substantial performance differences in long-term online serving. Fortunately, due to the strong commonsense knowledge and role-play capability of large language models (LLMs), it is possible to obtain simulated user feedback on offline recommendation results. Motivated by the idea of LLM Chatbot Arena, in this paper we present the idea of RecSys Arena, where the recommendation results given by two different recommender systems in each session are evaluated by an LLM judger to obtain fine-grained evaluation feedback. More specifically, for each sample we use LLM to generate a user profile description based on user behavior history or off-the-shelf profile features, which is used to guide LLM to play the role of this user and evaluate the relative preference for two recommendation results generated by different models. Through extensive experiments on two recommendation datasets in different scenarios, we demonstrate that many different LLMs not only provide general evaluation results that are highly consistent with canonical offline metrics, but also provide rich insight in many subjective aspects. Moreover, it can better distinguish different algorithms with comparable performance in terms of AUC and nDCG.

cs.IR↗

arXiv2024

Clique density vs blowups

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include: For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline{G}$ has positive triangle density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline{G}$ has positive $K_r$-density, then $\overline{G}$ contains a biclique of size $Ω(\frac{n}{\log{n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2. Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $Ω(\frac{n}{\log{n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $Ω(n)$, where the constant $(2h-2)^{r}+1$ is optimal. The $\frac{n}{\log n}$ size of the blowups in all our results are optimal up to a constant factor.

math.CO↗

arXiv2024

Empty red-red-blue triangles

Let $P$ be a $2n$-point set in the plane that is in general position. We prove that every red-blue bipartition of $P$ into $R$ and $B$ with $|R| = |B| = n$ generates $Ω(n^{3/2})$ red-red-blue empty triangles.

math.CO↗

arXiv2024

The uniform Turán density of large stars

We asymptotically resolve the the uniform Turán density problem for the large stars. In particular, we show that the uniform Turán density of the $k$-star $S_k$ is $\frac{k^2-5k+7}{(k-1)^2}$ for $k\ge 48$, matching a lower construction by Reiher, Rödl and Schacht.

math.CO↗

arXiv2023

Counting cliques without generalized theta graphs

The \textit{generalized Tur\'an number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection between $\ex(n,K_3,B_t)$ and the Ruzsa-Szemer\'edi triangle removal lemma. Motivated by this, in this paper, we study the generalized Tur\'an problem for generalized theta graphs, a natural extension of book graphs. Our main result provides a complete characterization of the magnitude of $\ex(n,K_3,H)$ when $H$ is a generalized theta graph, indicating when it is quadratic, when it is nearly quadratic, and when it is subquadratic. Furthermore, as an application, we obtain the exact value of $\ex(n, K_r, kF)$, where $F$ is an edge-critical generalized theta graph, and $3\le r\le k+1$, extending several recent results.

math.CO↗

arXiv2023

Distribution of colours in rainbow H-free colourings

An edge colouring of $K_n$ with $k$ colours is a Gallai $k$-colouring if it does not contain any rainbow triangle. Gyárfás, Pálvölgyi, Patkós and Wales proved that there exists a number $g(k)$ such that $n\geq g(k)$ if and only if for any colour distribution sequence $(e_1,\cdots,e_k)$ with $\sum_{i=1}^ke_i=\binom{n}{2}$, there exist a Gallai $k$-colouring of $K_n$ with $e_i$ edges having colour $i$. They also showed that $Ω(k)=g(k)=O(k^2)$ and posed the problem of determining the exact order of magnitude of $g(k)$. Feffer, Fu and Yan improved both bounds significantly by proving $Ω(k^{1.5}/\log k)=g(k)=O(k^{1.5})$. We resolve this problem by showing $g(k)=Θ(k^{1.5}/(\log k)^{0.5})$. Moreover, we generalise these definitions by considering rainbow $H$-free colourings of $K_n$ for any general graph $H$, and the natural corresponding quantity $g(H,k)$. We prove that $g(H,k)$ is finite for every $k$ if and only if $H$ is not a forest, and determine the order of $g(H,k)$ when $H$ contains a subgraph with minimum degree at least 3.

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arXiv2023

Revisiting Deep Neural Network Test Coverage from the Test Effectiveness Perspective

Many test coverage metrics have been proposed to measure the Deep Neural Network (DNN) testing effectiveness, including structural coverage and non-structural coverage. These test coverage metrics are proposed based on the fundamental assumption: they are correlated with test effectiveness. However, the fundamental assumption is still not validated sufficiently and reasonably, which brings question on the usefulness of DNN test coverage. This paper conducted a revisiting study on the existing DNN test coverage from the test effectiveness perspective, to effectively validate the fundamental assumption. Here, we carefully considered the diversity of subjects, three test effectiveness criteria, and both typical and state-of-the-art test coverage metrics. Different from all the existing studies that deliver negative conclusions on the usefulness of existing DNN test coverage, we identified some positive conclusions on their usefulness from the test effectiveness perspective. In particular, we found the complementary relationship between structural and non-structural coverage and identified the practical usage scenarios and promising research directions for these existing test coverage metrics.

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arXiv2023

Post-training Model Quantization Using GANs for Synthetic Data Generation

Quantization is a widely adopted technique for deep neural networks to reduce the memory and computational resources required. However, when quantized, most models would need a suitable calibration process to keep their performance intact, which requires data from the target domain, such as a fraction of the dataset used in model training and model validation (i.e. calibration dataset). In this study, we investigate the use of synthetic data as a substitute for the calibration with real data for the quantization method. We propose a data generation method based on Generative Adversarial Networks that are trained prior to the model quantization step. We compare the performance of models quantized using data generated by StyleGAN2-ADA and our pre-trained DiStyleGAN, with quantization using real data and an alternative data generation method based on fractal images. Overall, the results of our experiments demonstrate the potential of leveraging synthetic data for calibration during the quantization process. In our experiments, the percentage of accuracy degradation of the selected models was less than 0.6%, with our best performance achieved on MobileNetV2 (0.05%). The code is available at: https://github.com/ThanosM97/gsoc2022-openvino

cs.CV↗

arXiv2022

Positive co-degree Turán number for $C_5$ and $C_5^{-}$

The \emph{minimum positive co-degree} $δ^{+}_{r-1}(H)$ of a non-empty $r$-graph $H$ is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. For any $r$-graph $F$, the \emph{positive degree Turán number} $\mathrm{co}^{+}\mathrm{ex}(n,F)$ is defined as the maximum value of $δ^{+}_{r-1}(H)$ over all $n$-vertex $F$-free non-empty $r$-graphs $H$. In this paper, we determine the positive degree Turán number for $C_5$ and $C_5^{-}$.

math.CO↗

arXiv2022

Disjoint isomorphic balanced clique subdivisions

A thoroughly studied problem in Extremal Graph Theory is to find the best possible density condition in a host graph $G$ for guaranteeing the presence of a particular subgraph $H$ in $G$. One such classical result, due to Bollobás and Thomason, and independently Komlós and Szemerédi, states that average degree $O(k^2)$ guarantees the existence of a $K_k$-subdivision. We study two directions extending this result. On the one hand, Verstraëte conjectured that the quadratic bound $O(k^2)$ would guarantee already two vertex-disjoint isomorphic copies of a $K_k$-subdivision. On the other hand, Thomassen conjectured that for each $k \in \mathbb{N}$ there is some $d = d(k)$ such that every graph with average degree at least $d$ contains a balanced subdivision of $K_k$, that is, a copy of $K_k$ where the edges are replaced by paths of equal length. Recently, Liu and Montgomery confirmed Thomassen's conjecture, but the optimal bound on $d(k)$ remains open. In this paper, we show that the quadratic bound $O(k^2)$ suffices to force a balanced $K_k$-subdivision. This gives the optimal bound on $d(k)$ needed in Thomassen's conjecture and implies the existence of $O(1)$ many vertex-disjoint isomorphic $K_k$-subdivisions, confirming Verstraëte's conjecture in a strong sense.

math.CO↗

arXiv2021

On the stability of graph independence number

Let $G$ be a graph on $n$ vertices of independence number $\alpha(G)$ such that every induced subgraph of $G$ on $n-k$ vertices has an independent set of size at least $\alpha(G) - \ell$. What is the largest possible $\alpha(G)$ in terms of $n$ for fixed $k$ and $\ell$? We show that $\alpha(G) \le n/2 + C_{k, \ell}$, which is sharp for $k-\ell \le 2$. We also use this result to determine new values of the Erd\H{o}s--Rogers function.

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