arXiv subjects

Shumin Sun

Publications and source records attributed to Shumin Sun.

12 recordsLinked to original sources

arXiv2026

Strong non-principality of positive codegree Tur\'an density

The \emph{minimum positive codegree} $\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Tur\'an density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $\delta^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<\gamma^+(F_1, F_2) < \min\{\gamma^+(F_1), \gamma^+(F_2)\}. $$

math.CO↗

arXiv2026

The inducibility of 6-vertex graphs

The inducibility constant $\lambda_{F}$ of a graph $F$ is the asymptotically maximum induced density of $F$ in a growing sequence of graphs. This paper systematically investigates the case when $F$ has 6 vertices (and there are 78 cases to consider up to isomorphism and complementation). We show that flag algebras can compute the sharp upper bound on $\lambda_F$ in 36 cases of which, as far as the authors know, 30 are new results. In each of the solved cases, we also prove results about the structure of large (almost) extremal graphs. In particular, we establish perfect stability in all 32 cases when the extremal construction has no quasirandom parts. We also present conjectures about the value of $\lambda_{F}$ for 12 further cases (where the upper and lower bounds are very close to each other).

math.CO↗

arXiv2026

Rational codegree Tur\'an density of hypergraphs

Let $H$ be a $k$-graph (i.e. a $k$-uniform hypergraph). Its minimum codegree $\delta_{k-1}(H)$ is the largest integer $t$ such that every $(k-1)$-subset of $V(H)$ is contained in at least $t$ edges of~$H$. The \emph{codegree Tur\'an density} $\gamma(\mathcal{F})$ of a family $\mathcal{F}$ of $k$-graphs is the infimum of $\gamma > 0$ such that every $k$-graph $H$ on $n\to\infty$ vertices with $\delta_{k-1}(H) \ge (\gamma+o(1))\, n$ contains some member of $\mathcal{F}$ as a subgraph. We prove that, for every integer $k\ge3$ and every rational number $\alpha \in [0,1)$, there exists a finite family of $k$-graphs $\mathcal{F}$ such that $\gamma(\mathcal{F})=\alpha$. Also, for every $k \ge 3$, we establish a strong version of non-principality, namely that there are two $k$-graphs $F_1$ and $F_2$ such that the codegree Tur\'an density of $\{F_1,F_2\}$ is strictly smaller than that of each $F_i$. This answers a question of Mubayi and Zhao [J Comb Theory (A) 114 (2007) 1118--1132].

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arXiv2025

Covering large-dimensional Euclidean spaces by random translates of a given convex body

Determining the minimum density of a covering of $\mathbb{R}^{n}$ by Euclidean unit balls as $n\to\infty$ is a major open problem, with the best known results being the lower bound of $\left(\mathrm{e}^{-3/2}+o(1)\right)n$ by Coxeter, Few and Rogers [Mathematika 6, 1959] and the upper bound of $\left(1/2+o(1) \right)n \ln n$ by Dumer [Discrete Comput. Geom. 38, 2007]. We prove that there are ball coverings of $\mathbb{R}^n$ attaining the asymptotically best known density $\left(1/2+o(1) \right)n \ln n$ such that, additionally, every point of $\mathbb{R}^n$ is covered at most $\left(1.79556... + o(1)\right) n \ln n$ times. This strengthens the result of Erd\H{o}s and Rogers [Acta Arith. 7, 1961/62] who had the maximum multiplicity at most $\left(\mathrm{e} + o(1)\right) n \ln n$. On the other hand, we show that the method that was used for the best known ball coverings (when one takes a random subset of centres in a fundamental domain of a suitable lattice in $\mathbb{R}^n$ and extends this periodically) fails to work if the density is less than $(1/2+o(1))n\ln n$; in fact, this result remains true if we replace the ball by any convex body $K$. Also, we observe that a ``worst'' convex body $K$ here is a cube, for which the packing density coming from random constructions is only $(1+o(1))n\ln n$.

math.CO↗

arXiv2025

New upper bound for lattice covering by spheres

We show that there exists a lattice covering of $\mathbb{R}^n$ by Eucledian spheres of equal radius with density $O\big(n \ln^{\beta} n \big)$ as $n\to\infty$, where \begin{align*} \beta := \frac{1}{2} \log_2 \left(\frac{8 \pi \mathrm{e}}{3\sqrt 3}\right)=1.85837...\,. \end{align*} This improves upon the previously best known upper bound by Rogers from 1959 of $O\big(n \ln^{\alpha} n \big)$, where $\alpha := \frac{1}{2} \log_{2}(2\pi \mathrm{e})=2.0471...\,.$

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arXiv2025

On the quadratic 8-edge case of the Brown-Erd\H{o}s-S\'os problem

Let $f^{(r)}(n;s,k)$ be the maximum number of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ edges on at most $s$ vertices. Brown, Erd\H{o}s and S\'os conjectured in 1973 that the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(3)}(n;k+2,k)$ exists for all $k$. Recently, Delcourt and Postle settled the conjecture and their approach was generalised by Shangguan to every uniformity $r\ge 4$: the limit $\lim_{n\rightarrow \infty}n^{-2}f^{(r)}(n;rk-2k+2,k)$ exists for all $r\ge 3$ and $k\ge 2$. The value of the limit is currently known for $k\in \{2,3,4,5,6,7\}$ due to various results authored by Glock, Joos, Kim, K\"{u}hn, Lichev, Pikhurko, R\"odl and Sun. In this paper we consider the case $k=8$, determining the value of the limit for each $r\ge 4$ and presenting a lower bound for $k=3$ that we conjecture to be sharp.

math.CO↗

arXiv2024

Induced even cycles in locally sparse graphs

A graph $G$ is $(c,t)$-sparse if for every pair of vertex subsets $A,B\subset V(G)$ with $|A|,|B|\geq t$, $e(A,B)\leq (1-c)|A||B|$. In this paper we prove that for every $c>0$ and integer $\ell$, there exists $C>1$ such that if an $n$-vertex graph $G$ is $(c,t)$-sparse for some $t$, and has at least $C t^{1-1/\ell}n^{1+1/\ell}$ edges, then $G$ contains an induced copy of $C_{2\ell}$. This resolves a conjecture of Fox, Nenadov and Pham.

math.CO↗

arXiv2024

On the $(k+2,k)$-problem of Brown, Erd\H{o}s and S\'os for $k=5,6,7$

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, Erd\H{o}s and S\'os [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, Erd\H{o}s and S\'os, for $k=3$ by Glock [Bull. Lond. Math. Soc. 51 (2019) 230--236] and for $k=4$ by Glock, Joos, Kim, K\"uhn, 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-Erd\H{o}s-S\'os 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.

math.CO↗

arXiv2023

Tilings in quasi-random $k$-partite hypergraphs

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex disjoint copies of $F$ that together cover the vertex set of $H$. Lenz and Mubayi were first to study the $F$-factor problems in quasi-random $k$-graphs with a minimum degree condition. Recently, Ding, Han, Sun, Wang and Zhou gave the density threshold for having all $3$-partite $3$-graphs factors in quasi-random $3$-graphs with vanishing minimum codegree condition $\Omega(n)$. In this paper, we consider embedding factors when the host $k$-graph is $k$-partite and quasi-random with partite minimum codegree condition. We prove that if $p>1/2$ and $F$ is a $k$-partite $k$-graph with each part having $m$ vertices, then for $n$ large enough and $m\mid n$, any $p$-dense $k$-partite $k$-graph with each part having $n$ vertices and partite minimum codegree condition $\Omega(n)$ contains an $F$-factor. We also present a construction showing that $1/2$ is best possible. Furthermore, for $1\leq \ell \leq k-2$, by constructing a sequence of $p$-dense $k$-partite $k$-graphs with partite minimum $\ell$-degree $\Omega(n^{k-\ell})$ having no $K_k(m)$-factor, we show that the partite minimum codegree constraint can not be replaced by other partite minimum degree conditions. On the other hand, we prove that $n/2$ is the asymptotic partite minimum codegree threshold for having all fixed $k$-partite $k$-graph factors in sufficiently large host $k$-partite $k$-graphs even without quasi-randomness.

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arXiv2022

Optimal Allocation of Virtual Inertia and Droop Control for Renewable Energy in Stochastic Look-Ahead Power Dispatch

To stabilize the frequency of the renewable energy sources (RESs) dominated power system, frequency supports are required by RESs through virtual inertia emulation or droop control in the newly published grid codes. Since the long-term RES prediction involves significant errors, we need online configure the frequency control parameters of RESs in a rolling manner to improve the operation economics under the premise of stabilizing system frequency. To address this concern, this paper proposes a frequency constrained stochastic look-ahead power dispatch (FCS-LAPD) model to formulate the frequency control parameters of RESs and Energy Storage Systems (ESSs) as scheduling variables, which can optimally allocate the virtual inertia and droop coefficient of RESs and ESSs. In this FCS-LAPD model, the uncertainties of RESs are characterized using Gaussian Mixture Model (GMM). The required reserves are determined by frequency control parameters, and the reserve cost coefficients are adjusted properly to allocate the reserves according to the predicted power generation. Due to the nonlinearity of the frequency nadir constraint, a convex hull approximation method is proposed to linearize it with guaranteed feasibility. The proposed FCS-LAPD is ultimately cast as an instance of quadratic programming and can be efficiently solved. Case studies on modified IEEE 24-bus system and a provincial power system in China are conducted to show the effectiveness of the proposed model.

eess.SY↗

arXiv2021

Tiling multipartite hypergraphs in Quasi-random Hypergraphs

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $\Omega(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $\Omega(n^2)$ and minimum codegree $\Omega(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense quasi-random $3$-graphs of order $n$ with minimum codegree $\Omega(n)$ and $f\mid n$ have $F$-factors. That is, $1/8$ is the density threshold for ensuring all $3$-partite $3$-graphs $F$-factors in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$. Moreover, we show that one can not replace the minimum codegree condition by a minimum vertex degree condition. In fact, we find that for any $p\in(0,1)$ and $n\ge n_0$, there exist $p$-dense quasi-random $3$-graphs of order $n$ with minimum degree $\Omega (n^2)$ having no $K_{2,2,2}$-factor. In particular, we study the optimal density threshold of $F$-factors for each $3$-partite $3$-graph $F$ in quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$.

math.CO↗

arXiv2021

$F$-factors in Quasi-random Hypergraphs

Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an $F$-factor in $H$ is a set of vertex-disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the $F$-factor problem in quasi-random $k$-graphs with minimum degree $\Omega(n^{k-1})$. They posed the problem of characterizing the $k$-graphs $F$ such that every sufficiently large quasi-random $k$-graph with constant edge density and minimum degree $\Omega(n^{k-1})$ contains an $F$-factor, and in particular, they showed that all linear $k$-graphs satisfy this property. In this paper we prove a general theorem on $F$-factors which reduces the $F$-factor problem of Lenz and Mubayi to a natural sub-problem, that is, the $F$-cover problem. By using this result, we answer the question of Lenz and Mubayi for those $F$ which are $k$-partite $k$-graphs, and for all 3-graphs $F$, separately. Our characterization result on 3-graphs is motivated by the recent work of Reiher, R\"odl and Schacht [J. Lond. Math. Soc., 2018] that classifies the 3-graphs with vanishing Tur\'an density in quasi-random $k$-graphs.

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