arXiv subjects
Publications and source records attributed to Jaehoon Kim.
Determining the maximum number of edges in an intersecting hypergraph on a fixed ground set under additional constraints is one of the central topics in extremal combinatorics. In contrast, there are few results on analogous problems concerning the maximum order of such hypergraphs. In this paper, we systematically study these vertex analogues.
Let $f^{(r)}(n;s,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no subgraph with $k$ edges and at most $s$ vertices. Brown, Erdős and Sós [New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan 1971), pp. 53--63, Academic Press 1973] conjectured that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. The value of the limit was previously determined for $k=2$ in the original paper of Brown, Erdős and Sós, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, Kühn, Lichev and Pikhurko [Proc. Amer. Math. Soc., Series B, 11 (2024) 173-186] while Delcourt and Postle [Proc. Amer. Math. Soc., 152 (2024), 1881-1891] proved the conjecture (without determining the limiting value). In this paper, we determine the value of the limit in the Brown-Erdős-Sós Problem for $k\in \{5,6,7\}$. More generally, we obtain the value of $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ for all $r\geq 3$ and $k\in \{5,6,7\}$. In addition, by combining these new values with recent results of Bennett, Cushman and Dudek [arXiv:2309.00182] we obtain new asymptotic values for several generalised Ramsey numbers.
Given a graph $H$, a balanced subdivision of $H$ is obtained by replacing all edges of $H$ with internally disjoint paths of the same length. In this paper, we prove that for any graph $H$, a linear-in-$e(H)$ bound on average degree guarantees a balanced $H$-subdivision. This strengthens an old result of Bollobás and Thomason, and resolves a question of Gil-Fernández, Hyde, Liu, Pikhurko and Wu. We observe that this linear bound on average degree is best possible whenever $H$ is logarithmically dense. We further show that this logarithmic density is the critical threshold: for many graphs $H$ below this density, its subdivisions are forcible by a sublinear-in-$e(H)$ bound on average degree. We provide such examples by proving that the subdivisions of any almost bipartite graph $H$ with sublogarithmic density are forcible by a sublinear-in-$e(H)$ bound on average degree, provided that $H$ satisfies some additional separability condition.
Ramsey--Turán theory considers Turán type questions in Ramsey-context, asking for the existence of a small subgraph in a graph $G$ where the complement $\overline{G}$ lacks an appropriate subgraph $F$, such as a clique of linear size. Similarly, one can consider Dirac-type questions in Ramsey context, asking for the existence of a spanning subgraph $H$ in a graph $G$ where the complement $\overline{G}$ lacks an appropriate subgraph $F$, which we call a Ramsey--Dirac theory question. When $H$ is a connected spanning subgraph, the disjoint union $K_{n/2}\cup K_{n/2}$ of two large cliques shows that it is natural to consider complete bipartite graphs $F$. Indeed, Han, Hu, Ping, Wang, Wang and Yang in 2024 proved that if $G$ is an $n$-vertex graph with $δ(G)=Ω(n)$ where the complement $\overline{G}$ does not contain any complete bipartite graph $K_{m,m}$ with $m=Ω(n)$, then $G$ contains every $n$-vertex bounded degree tree $T$ as a subgraph. Extending this result to the Ramsey--Dirac theory for hypertrees, we prove that if $G$ is an $n$-vertex $r$-uniform hypergraph with $δ(G)=Ω(n^{r-1})$ where the complement $\overline{G}$ does not contain any complete $r$-partite hypergraph $K_{m,m,\dots, m}$ with $m=Ω(n)$, then $G$ contains every $n$-vertex bounded degree hypertree $T$ as a subgraph. We also prove the existence of matchings and loose Hamilton cycles in the same setting, which extends the result of Mcdiarmid and Yolov into hypergraphs. This result generalizes the universality result on randomly perturbed graphs by Böttcher, Han, Kohayakawa, Montgomery, Parczyk and Person in 2019 into hypergraphs and also strengthen the results on quasirandom hypergraphs by Lenz, Mubayi and Mycroft in 2016 and Lenz and Mubayi in 2016 into hypergraphs satisfying a much weaker pseudorandomness condition.
We propose the first Bayesian methods for detecting change points in high-dimensional mean and covariance structures. These methods are constructed using pairwise Bayes factors, leveraging modularization to identify significant changes in individual components efficiently. We establish that the proposed methods consistently detect and estimate change points under much milder conditions than existing approaches in the literature. Additionally, we demonstrate that their localization rates are nearly optimal in terms of rates. The practical performance of the proposed methods is evaluated through extensive simulation studies, where they are compared to state-of-the-art techniques. The results show comparable or superior performance across most scenarios. Notably, the methods effectively detect change points whenever signals of sufficient magnitude are present, irrespective of the number of signals. Finally, we apply the proposed methods to genetic and financial datasets, illustrating their practical utility in real-world applications.
The famous Dirac's Theorem states that for each $n\geq 3$ every $n$-vertex graph $G$ with minimum degree $δ(G)\geq n/2$ has a hamiltonian cycle. When $δ(G)< n/2$, this cannot be guaranteed, but the existence of some other specific subgraphs can be provided. Gargano, Hell, Stacho and Vaccaro proved that every connected $n$-vertex graph $G$ with $δ(G)\geq (n-1)/3$ contains a spanning {\em spider}, i.e., a spanning tree with at most one vertex of degree at least $3$. Later, Chen, Ferrara, Hu, Jacobson and Liu proved the stronger (and exact) result that for $n\geq 56$ every connected $n$-vertex graph $G$ with $δ(G)\geq (n-2)/3$ contains a spanning {\em broom}, i.e., a spanning spider obtained by joining the center of a star to an endpoint of a path. They also showed that a $2$-connected graph $G$ with $δ(G)\geq (n-2)/3$ and some additional properties contains a spanning {\em jellyfish} which is a graph obtained by gluing the center of a star to a vertex in a cycle disjoint from that star. Note that every spanning jellyfish contains a spanning broom. The goal of this paper is to prove an exact Ore-type bound which guarantees the existence of a spanning jellyfish: We prove that if $G$ is a $2$-connected graph on $n$ vertices such that every non-adjacent pair of vertices $(u,v)$ satisfies $d(u) + d(v) \geq \frac{2n-3}{3}$, then $G$ has a spanning jellyfish. As corollaries, we obtain strengthenings of two results by Chen et al.: a minimum degree condition guaranteeing the existence of a spanning jellyfish, and an Ore-type sufficient condition for the existence of a spanning broom. The corollaries are sharp for infinitely many $n$. One of the main ingredients of our proof is a modification of the Hopping Lemma due to Woodall.
We present a new lower bound on the Bowen-Radin maximal density of radius-R ball packings in the m-dimensional hyperbolic space, improving on the basic covering bound by factor Ω(m(R+\ln m)) as m tends to infinity. This is done by applying the recent theorem of Campos, Jenssen, Michelen and Sahasrabudhe on independent sets in graphs with sparse neighbourhoods.
Machine learning interatomic potentials (MLIPs) are used to estimate potential energy surfaces (PES) from ab initio calculations, providing near quantum-level accuracy with reduced computational costs. However, the high cost of assembling high-fidelity databases hampers the application of MLIPs to systems that require high chemical accuracy. Utilizing an equivariant graph neural network, we present an MLIP framework that trains on multi-fidelity databases simultaneously. This approach enables the accurate learning of high-fidelity PES with minimal high-fidelity data. We test this framework on the Li$_6$PS$_5$Cl and In$_x$Ga$_{1-x}$N systems. The computational results indicate that geometric and compositional spaces not covered by the high-fidelity meta-gradient generalized approximation (meta-GGA) database can be effectively inferred from low-fidelity GGA data, thus enhancing accuracy and molecular dynamics stability. We also develop a general-purpose MLIP that utilizes both GGA and meta-GGA data from the Materials Project, significantly enhancing MLIP performance for high-accuracy tasks such as predicting energies above hull for crystals in general. Furthermore, we demonstrate that the present multi-fidelity learning is more effective than transfer learning or $Δ$-learning an d that it can also be applied to learn higher-fidelity up to the coupled-cluster level. We believe this methodology holds promise for creating highly accurate bespoke or universal MLIPs by effectively expanding the high-fidelity dataset.
Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This paper establishes $\textit{transversal}$ generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on the common vertex set $V$, an $m$-edge directed subgraph $\mathcal{D}$ with the vertices in $V$ is called a transversal if there exists an bijection $φ\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{φ(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $n$, there exist transversal Hamilton cycles of all possible orientations possibly except the consistently oriented one. We also obtain a similar result for the transversal Hamilton paths of all possible orientations. These results generalize the classical theorem of Thomason, and our approach provides another proof of this theorem.
Detecting implicit hate speech that is not directly hateful remains a challenge. Recent research has attempted to detect implicit hate speech by applying contrastive learning to pre-trained language models such as BERT and RoBERTa, but the proposed models still do not have a significant advantage over cross-entropy loss-based learning. We found that contrastive learning based on randomly sampled batch data does not encourage the model to learn hard negative samples. In this work, we propose Label-aware Hard Negative sampling strategies (LAHN) that encourage the model to learn detailed features from hard negative samples, instead of naive negative samples in random batch, using momentum-integrated contrastive learning. LAHN outperforms the existing models for implicit hate speech detection both in- and cross-datasets. The code is available at https://github.com/Hanyang-HCC-Lab/LAHN
It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on a common vertex set $V$, an $m$-edge directed graph $\mathcal{D}$ with vertices in $V$ is called a $\mathbf{T}$-transversal if there exists a bijection $ϕ\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{ϕ(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $m$ with $m=|V|-1$, there exists a $\mathbf{T}$-transversal Hamilton path. Moreover, if $m=|V|$ and at least $m-1$ of the tournaments $T_1,\ldots,T_m$ are assumed to be strongly connected, then there is a $\mathbf{T}$-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub $\mathbf{H}$-partition.
Liu, Pach and Sándor recently characterized all polynomials $p(z)$ such that the equation $x+y=p(z)$ is $2$-Ramsey, that is, any $2$-coloring of $\mathbb{N}$ contains infinitely many monochromatic solutions for $x+y=p(z)$. In this paper, we find asymptotically tight bounds for the following two quantitative questions. $\bullet$ For $n\in \mathbb{N}$, what is the longest interval $[n,f(n)]$ of natural numbers which admits a $2$-coloring with no monochromatic solutions of $x+y=p(z)$? $\bullet$ For $n\in \mathbb{N}$ and a $2$-coloring of the first $n$ integers $[n]$, what is the smallest possible number $g(n)$ of monochromatic solutions of $x+y=p(z)$? Our theorems determine $f(n)$ up to a multiplicative constant $2+o(1)$, and determine the asymptotics for $g(n)$.
Considerable efforts are currently underway to mitigate the negative impacts of echo chambers, such as increased susceptibility to fake news and resistance towards accepting scientific evidence. Prior research has presented the development of computer systems that support the consumption of news information from diverse political perspectives to mitigate the echo chamber effect. However, existing studies still lack the ability to effectively support the key processes of news information consumption and quantitatively identify a political stance towards the information. In this paper, we present HearHere, an AI-based web system designed to help users accommodate information and opinions from diverse perspectives. HearHere facilitates the key processes of news information consumption through two visualizations. Visualization 1 provides political news with quantitative political stance information, derived from our graph-based political classification model, and users can experience diverse perspectives (Hear). Visualization 2 allows users to express their opinions on specific political issues in a comment form and observe the position of their own opinions relative to pro-liberal and pro-conservative comments presented on a map interface (Here). Through a user study with 94 participants, we demonstrate the feasibility of HearHere in supporting the consumption of information from various perspectives. Our findings highlight the importance of providing political stance information and quantifying users' political status as a means to mitigate political polarization. In addition, we propose design implications for system development, including the consideration of demographics such as political interest and providing users with initiatives.
In 1994, Alon construct a triangle-free $(n,d,λ)$-graph with $d = Ω(n^{2/3})$ and $λ= O(d^{1/2})$ for an exponentially increasing sequence of integers $n$. Using his ingenious construction, we deduce that there exist triangle-free $(n,d,λ)$-graphs with $d = Ω(n^{2/3})$ and $λ= O( (d \log n)^{1/2} )$ for all sufficiently large $n$.
For a given collection $\mathcal{G} = (G_1,\dots, G_k)$ of graphs on a common vertex set $V$, which we call a \emph{graph system}, a graph $H$ on a vertex set $V(H) \subseteq V$ is called a \emph{rainbow subgraph} of $\mathcal{G}$ if there exists an injective function $ψ:E(H) \rightarrow [k]$ such that $e \in G_{ψ(e)}$ for each $e\in E(H)$. The maximum value of $\min_{i}\{|E(G_i)|\}$ over $n$-vertex graph systems $\mathcal{G}$ having no rainbow subgraph isomorphic to $H$ is called the rainbow Turán number $\mathrm{ex}_k^{\ast}(n, H)$ of $H$. In this article, we study the rainbow Turán density $π_k^{\ast}(T) = \lim_{n \rightarrow \infty} \frac{\mathrm{ex}_k^{\ast}(n, T)}{\binom{n}{2}}$ of a tree $T$. While the classical Turán density $π(H)$ of a graph $H$ lies in the set $\{1-\frac{1}{t} : t\in \mathbb{N}\}$, the rainbow Turán density exhibits different behaviors as it can even be an irrational number. Nevertheless, we conjecture that the rainbow Turán density is always an algebraic number. We provide evidence for this conjecture by proving that the rainbow Turán density of a tree is an algebraic number. To show this, we identify the structure of extremal graphs for rainbow trees. Moreover, we further determine all tuples $(α_1,\dots, α_k)$ such that every graph system $(G_1,\dots,G_k)$ satisfying $|E(G_i)|>(α_i+o(1))\binom{n}{2}$ contains all rainbow $k$-edge trees. In the course of proving these results, we also develop the theory on the limit of graph systems.
Let the kissing number $K(d)$ be the maximum number of non-overlapping unit balls in $\mathbb R^d$ that can touch a given unit ball. Determining or estimating the number $K(d)$ has a long history, with the value of $K(3)$ being the subject of a famous discussion between Gregory and Newton in 1694. We prove that, as the dimension $d$ goes to infinity, $$ K(d)\ge (1+o(1)){\frac{\sqrt{3π}}{4\sqrt2}}\,\log\frac{3}{2}\cdot d^{3/2}\cdot \Big(\frac{2}{\sqrt{3}}\Big)^{d}, $$ thus improving the previously best known bound of Jenssen, Joos and Perkins by a factor of $\log(3/2)/\log(9/8)+o(1)=3.442...$. Our proof is based on the novel approach from Jenssen, Joos and Perkins that uses the hard core sphere model of an appropriate fugacity. Similar constant-factor improvements in lower bounds are also obtained for general spherical codes, as well as for the expected density of random sphere packings in the Euclidean space $\mathbb R^d$.
For a collection $\mathbf{G}=\{G_1,\dots, G_s\}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $\mathbf{G}$-transversal if there exists a bijection $ϕ:E(H)\rightarrow [s]$ such that $e\in E(G_{ϕ(e)})$ for all $e\in E(H)$. We prove that for $|V|=s\geq 3$ and $δ(G_i)\geq s/2$ for each $i\in [s]$, there exists a $\mathbf{G}$-transversal that is a Hamilton cycle. This confirms a conjecture of Aharoni. We also prove an analogous result for perfect matchings.
For a given graph $H$, its subdivisions carry the same topological structure. The existence of $H$-subdivisions within a graph $G$ has deep connections with topological, structural and extremal properties of $G$. One prominent example of such a connection, due to Bollobás and Thomason and independently Komlós and Szemerédi, asserts that the average degree of $G$ being $d$ ensures a $K_{Ω(\sqrt{d})}$-subdivision in $G$. Although this square-root bound is best possible, various results showed that much larger clique subdivisions can be found in a graph for many natural classes. We investigate the connection between crux, a notion capturing the essential order of a graph, and the existence of large clique subdivisions. This reveals the unifying cause underpinning all those improvements for various classes of graphs studied. Roughly speaking, when embedding subdivisions, natural space constraints arise; and such space constraints can be measured via crux. Our main result gives an asymptotically optimal bound on the size of a largest clique subdivision in a generic graph $G$, which is determined by both its average degree and its crux size. As corollaries, we obtain (1) a characterisation of extremal graphs for which the square-root bound above is tight: they are essentially disjoint unions of graphs having crux size linear in $d$; (2) a unifying approach to find a clique subdivision of almost optimal size in graphs which do not contain a fixed bipartite graph as a subgraph; (3) and that the clique subdivision size in random graphs $G(n,p)$ witnesses a dichotomy: when $p = ω(n^{-1/2})$, the barrier is the space, while when $p=o( n^{-1/2})$, the bottleneck is the density.