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827 results on '"Kim, Jaehoon"'

11. On a rainbow extremal problem for color-critical graphs

Author

Chakraborti, Debsoumya, Kim, Jaehoon, Lee, Hyunwoo, Liu, Hong, and Seo, Jaehyeon

Subjects
Mathematics - Combinatorics, 05C35, 05D99
Abstract

There has been extensive studies on the following question: given $k$ graphs $G_1,\dots, G_k$ over a common vertex set of size $n$, what conditions on $G_i$ ensures a `colorful' copy of $H$, i.e., a copy of $H$ containing at most one edge from each $G_i$? A lower bound on $\sum_{i\in [k]} e(G_i)$ enforcing a colorful copy of a given graph $H$ was considered by Keevash, Saks, Sudakov, and Verstra\"{e}te. They defined $\operatorname{ex}_k(n,H)$ to be the maximum total number of edges of the graphs $G_1,\dots, G_k$ on a common vertex set of size $n$ having no colorful copy of $H$. They completely determined $\operatorname{ex}_k(n,K_r)$ for large $n$ by showing that, depending on the value of $k$, one of the two natural constructions is always the extremal construction. Moreover, they conjectured the same holds for every color-critical graphs and proved it for 3-color-critical graphs. We prove their conjecture for 4-color-critical graphs and for almost all $r$-color-critical graphs when $r > 4$. Moreover, we show that for every non-color-critical non-bipartite graphs, none of the two natural constructions is extremal for certain values of $k$. This answers a question of Keevash, Saks, Sudakov, and Verstra\"{e}te.

Published
2022
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12. Bibliographic and Text Analysis of Research on Implementation of the Internet of Things to Support Education

Author

Desai, Roopa Ramesh and Kim, Jaehoon

Abstract

The Internet of Things (IoT) has pervaded practically all aspects of our lives. In this exploratory study, we survey its applications in the field of education. It is evident that technology in general, and, in particular IoT, has been increasingly altering the educational landscape. The goal of this paper is to review the academic literature on IoT applications in education to provide an understanding of the transformation that is underway. Using topic modeling and keyword co-occurrence analysis techniques, we identified five dominant clusters of research. Our findings demonstrate that IoT research in education has mainly focused on the technical aspects; however, the social aspects remain largely unexplored. In addition to providing an overview of IoT research on education, this paper offers suggestions for future research.

Published
2023

22. Ramsey numbers of cycles versus general graphs

Author

Haslegrave, John, Hyde, Joseph, Kim, Jaehoon, and Liu, Hong

Subjects
Mathematics - Combinatorics, 05C55 (Primary) 05C38, 05C48 (Secondary)
Abstract

The Ramsey number $R(F,H)$ is the minimum number $N$ such that any $N$-vertex graph either contains a copy of $F$ or its complement contains $H$. Burr in 1981 proved a pleasingly general result that for any graph $H$, provided $n$ is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: $R(C_n,H)=(n-1)(\chi(H)-1)+\sigma(H)$, where $\sigma(H)$ is the minimum possible size of a colour class in a $\chi(H)$-colouring of $H$. Allen, Brightwell and Skokan conjectured that the same should be true already when $n\geq |H|\chi(H)$. We improve this 40-year-old result of Burr by giving quantitative bounds of the form $n\geq C|H|\log^4\chi(H)$, which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs $H$ with large chromatic number., Comment: 20 pages, 3 figures. Final version to appear in Forum of Mathematics, Sigma

Published
2021
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27. Well-mixing vertices and almost expanders

Author

Chakraborti, Debsoumya, Kim, Jaehoon, Kim, Jinha, Kim, Minki, and Liu, Hong

Subjects
Mathematics - Combinatorics, Computer Science - Discrete Mathematics
Abstract

We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As a corollary, it shows that sparse (constant degree) regular graphs with many well-mixing vertices have a long cycle, improving a result of Pak. Furthermore, such cycle can be found in polynomial time. Secondly, we show that if the random walks from a positive fraction of vertices are well-mixing, then the random walks from almost all vertices are well-mixing (with a slightly worse mixing time)., Comment: accepted in PAMS

Published
2021
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28. Fractional Helly theorem for Cartesian products of convex sets

Author

Chakraborti, Debsoumya, Kim, Jaehoon, Kim, Jinha, Kim, Minki, and Liu, Hong

Subjects
Mathematics - Combinatorics, 05E45, 52A35
Abstract

Helly's theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-aligned boxes, which are Cartesian products of line segments. Answering a question raised by B\'ar\'any and Kalai, and independently Lew, we generalize Eckhoff's result to Cartesian products of convex sets in all dimensions. In particular, we prove that given $\alpha \in (1-\frac{1}{t^d},1]$ and a finite family $\mathcal{F}$ of Cartesian products of convex sets $\prod_{i\in[t]}A_i$ in $\mathbb{R}^{td}$ with $A_i\subset \mathbb{R}^d$ if at least $\alpha$-fraction of the $(d+1)$-tuples in $\mathcal{F}$ are intersecting then at least $(1-(t^d(1-\alpha))^{1/(d+1)})$-fraction of sets in $\mathcal{F}$ are intersecting. This is a special case of a more general result on intersections of $d$-Leray complexes. We also provide a construction showing that our result on $d$-Leray complexes is optimal. Interestingly the extremal example is representable as a family of cartesian products of convex sets, implying the bound $\alpha>1-\frac{1}{t^d}$ and the fraction $(1-(t^d(1-\alpha))^{1/(d+1)})$ above are also best possible. The well-known optimal construction for fractional Helly theorem for convex sets in $\mathbb{R}^d$ does not have $(p,d+1)$-condition for sublinear $p$. Inspired by this we give constructions showing that, somewhat surprisingly, imposing additional $(p,d+1)$-condition has negligible effect on improving the quantitative bounds in neither the fractional Helly theorem for convex sets nor Cartesian products of convex sets. Our constructions offer a rich family of distinct extremal configurations for fractional Helly theorem, implying in a sense that the optimal bound is stable., Comment: 17 pages, 2 figures

Published
2021
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31. MLL-AF4 cooperates with PAF1 and FACT to drive high-density enhancer interactions in leukemia

Author

Crump, Nicholas T., Smith, Alastair L., Godfrey, Laura, Dopico-Fernandez, Ana M., Denny, Nicholas, Harman, Joe R., Hamley, Joseph C., Jackson, Nicole E., Chahrour, Catherine, Riva, Simone, Rice, Siobhan, Kim, Jaehoon, Basrur, Venkatesha, Fermin, Damian, Elenitoba-Johnson, Kojo, Roeder, Robert G., Allis, C. David, Roberts, Irene, Roy, Anindita, Geng, Huimin, Davies, James O. J., and Milne, Thomas A.

Published
2023
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32. Mitochondrial matrix protein LETMD1 maintains thermogenic capacity of brown adipose tissue in male mice

Author

Park, Anna, Kim, Kwang-eun, Park, Isaac, Lee, Sang Heon, Park, Kun-Young, Jung, Minkyo, Li, Xiaoxu, Sleiman, Maroun Bou, Lee, Su Jeong, Kim, Dae-Soo, Kim, Jaehoon, Lim, Dae-Sik, Woo, Eui-Jeon, Lee, Eun Woo, Han, Baek Soo, Oh, Kyoung-Jin, Lee, Sang Chul, Auwerx, Johan, Mun, Ji Young, Rhee, Hyun-Woo, Kim, Won Kon, Bae, Kwang-Hee, and Suh, Jae Myoung

Published
2023
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