Nikita Dubinin Andreevich
In this paper, we consider the Johnson's graphs. We study the extremal properties of the Johnson's graphs. Namely, we investigate the number of edges in an arbitrary subgraph of this graph. Namely, in this article we prove analogs of Turan's 1941 theorem.
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The generalized Turán number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of a graph $G.$ It is an important extension of the classical Turán number ${\rm ex}(n,H)$, which is the maximum number of edges in a graph with $n$ vertices that does not contain $H$ as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs $J(n;k,k+1)$, which are bipartite subgraphs of hypercube graphs. We give an upper bound for ${\rm ex}(J(n;k,k+1),C_{2r})$ with any fixed $k\in\mathbb{Z}^+$ and any $n\in\mathbb{Z}^+$ wi…
Mutual-visibility problems in Kneser and Johnson graphs
本文研究连通图中的互见性问题，确定了Kneser图、二部Kneser图和Johnson图的全互见数，并将其与横切临界均匀超图规模及超图Turán数建立联系，同时给出了这些图类互见数的精确值或估计。
An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Turán number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Turán numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraëte. Johnson and Rombach introduced the following rainbow-version of generalized Turán problems: for fixed graphs $H$ and $F$, let $\mathrm{ex}^*(n,H,F)$ denote the maximum number of rainbow copies of $H$ in an $n$-vertex properly edge-colored gra…
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The Turán number $\text{ex}(n,\mathcal{H})$ is the maximum number of edges in an $n$-vertex $\mathcal{H}$-free $r$-uniform hypergraph. The Turán density of $\mathcal{H}$ is defined by \[π(\mathcal{H})=\lim_{n\rightarrow\infty}\frac{\text{ex}(n,\mathcal{H})}{\binom{n}{r}}.\] In this paper, we consider the Turán density of projective geometries. We give two new constructions of $PG_{m}(q)$-free hypergraphs which improve some results given by Keevash (J. Combin. Theory Ser. A, 111: 289--309, 2005). Based on an upper bound of blocking sets of $PG_m(q…
Turán number is one of primary topics in the combinatorics of finite sets,in this paper, we will present a new upper bound for Turán number.
On generalized Turán problems with bounded matching number
本文研究了具有受限匹配数的广义Turán问题，确立了相关的稳定性结果，并利用这些结果给出了当禁用图F为任意非二部图或k顶点路径时广义Turán数的精确值。
Upper Bounds on Turán Densities via Extremal Set Theory
本文系统展示了超图 Turán 问题与极值集合论之间的联系，通过构造一致超图族，将其 Turán 密度的上界归结为极值集合论中的经典问题。
We consider in detail the well-known family of graphs $G(q,t)$ that establish an asymptotic lower bound for Turán numbers $\mathrm{ex}(n,K_{2,t+1})$. We prove that $G(q,t)$ for some specific $q$ and $t$ also gives an asymptotic bound for $K_{3,3}$ and for some higher complete bipartite graphs as well. The asymptotic bounds we prove are the same as provided by the well-known Norm-graphs.
In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many other examples of this density. Fon-der-Flaass (1988) presented a general construction that converts an arbitrary $\vec C_4$-free orgraph $Γ$ into a Turan (3,4)-graph. He observed that all Turan-Brown-Kostochka examples result from his construction, and proved the bound >= 3/7(1-o(1)) on the edge density of any Turan (3,4)-graph obtainable in this way. In th…
Localization of spectral Turán theorems for signed graphs
本文将局部Turán定理推广到符号图，并研究相应的谱Turán问题，建立了顶点局部Turán型不等式、最大特征值的尖锐上界以及基于游走的局部谱Turán不等式。
A linear $3$-graph is a set of vertices along with a set of edges, which are three element subsets of the vertices, such that any two edges intersect in at most one vertex. The crown, $C$, is a specific $3$-graph consisting of three pairwise disjoint edges, called jewels, along with a fourth edge intersecting all three jewels. For a linear $3$-graph, $F$, the linear Turán number, $ex(n,F)$, is the maximum number of edges in any linear $3$-graph that does not contain $F$ as a subgraph. Currently, the best known bounds on the linear Turán number of the crown are \[ 6 \Big \lfloor \frac{n-3}{4}\B…
On the Ramsey-Turán problem for 4-cliques
本文在不使用正则性引理的条件下，给出了4-团Ramsey-Turán问题的紧确界，从而大幅扩展了无$K_4$图边数关于独立数函数的紧确界成立范围。
In this note we obtain upper bounds on the number of hyperedges in 3-uniform hypergraphs not containing a Berge cycle of given odd length. We improve the bound given by Füredi and Özkahya in 2017. The result follows from a more general theorem. We also obtain some new results for Berge cliques.
Triple systems with bounded matching number: some constructions and exact Turán number
本文推翻了关于避免特定3-图和匹配的图论数的猜想，并确定了某些3-图的精确图论数。
For given graphs $G$ and $F$, the Turán number $ex(G,F)$ is defined to be the maximum number of edges in an $F$-free subgraph of $G$. Foucaud, Krivelevich and Perarnau and later independently Briggs and Cox introduced a dual version of this problem wherein for a given number $k$, one maximizes the number of edges in a host graph $G$ for which $ex(G,H) < k$. Addressing a problem of Briggs and Cox, we determine the asymptotic value of the inverse Turán number of the paths of length $4$ and $5$ and provide an improved lower bound for all paths of even length. Moreover, we obtain bounds on the inv…
For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a rainbow copy of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n, F)$, is the rainbow Turán number of $F$. We show that $ex^*(n,P_k)\geq \frac{k}{2}n + O(1)$ where $P_k$ is a path on $k\geq 3$ edges, generalizing a result by Maamoun and Meyniel and by Johnston, Palmer and Sarkar. We show similar bounds for brooms on $2^s-1$ edges and diameter $\leq 10$ and a few other caterpillars of sma…
Subspace codes, i.e., sets of subspaces of $\mathbb{F}_q^v$, are applied in random linear network coding. Here we give improved upper bounds for their cardinalities based on the Johnson bound for constant dimension codes.
The rainbow Turán number, a natural extension of the well studied traditional Turán number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraëte. The rainbow Turán number of a graph $H$, $ex^{*}(n,H)$, is the largest number of edges for an $n$ vertex graph $G$ which can be properly edge colored with no rainbow $H$ subgraph. We explore the reduction method for finding upper bounds on rainbow Turán numbers, and use this to inform results for the rainbow Turán numbers of double stars, caterpillars, and perfect binary trees. In addition, we define $k$-unique colorings and the related …
On generalized Turán problems with bounded matching number
本文将广义Turán问题中禁止$F$与匹配$M_{s+1}$且$H=K_r$的研究推广到$H$为任意图的情形，扩展了有界匹配数条件下广义Turán数的研究范围。
Let $F$ be a strictly balanced $r$-uniform hypergraph with $e>2$ edges and $r$-density $m$. We give a new short proof of the fact that the Turán number $\ex(n, F)$ is greater than $c\, n^{r-1/m} (\log n)^{1/(e-1)}$ where $c$ depends only on $F$. The previous proof of this for $r=2$ by Bohman and Keevash and for $r \ge 3$ by Bennett and Bohman used a random greedy process and its analysis using the differential equations method. Our proof uses elementary probabilistic arguments together with a (nontrivial) classical result about independent sets in hypergraphs.
Improved Sidorenko bound for hypergraph Turan densities
该研究针对r≥3的r-均匀超图，改进了Turán密度的西多连科界，将差距量级从m^(-1)缩小至m^(-(r-1)/r)，并通过完全r-均匀超图验证了指数的最优性，证明采用了部分嵌入的势论证与Lovász局部引理。
We consider the Turán-type problem of bounding the size of a set $M \subseteq \mathbb{F}_2^n$ that does not contain a linear copy of a given fixed set $N \subseteq \mathbb{F}_2^k$, where $n$ is large compared to $k$. An Erdős-Stone type theorem [5] in this setting gives a bound that is tight up to a $o(2^n)$ error term; our first main result gives a stability version of this theorem, showing that such an $M$ that is close in size to the upper bound in [5] is close in edit distance to the obvious extremal example. Our second result shows that the error term in [5] is exactly controlled by the s…
An Improved Turán Exponent for 2-Complexes
本文改进了二维复形的图论Turán指数，证明其最大边数不超过Cn^{8/3}，并提供了拓扑Turán数的紧上界新证明。
The planar Turan number $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ is the largest number of edges in an $n$-vertex planar graph with no $\ell$-cycle. For $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known that hold with equality infinitely often. Ghosh, Györi, Martin, Paulo, and Xiao [arxiv:2004.14094] conjectured an upper bound on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ for every $\ell\ge 7$ and $n$ sufficiently large. We disprove this conjecture for every $\ell\ge 11$. We also propose two revised versions of the conjecture.
Spectrum of Johnson graphs
本文为Johnson图第二特征值建立新界，并应用于其模块性、哈密顿性及随机子图阈值，同时给出$(n,d,\lambda)$-图的连通性与模块性稳定性界。
We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.
We propose a strengthening of the conclusion in Turán's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Turán theorem by weakening its assumption: for any graph G on n vertices and any involution on its vertex set, if for any 3-set S of the vertices, the number of edges in G spanned by S, plus the number of edges in G spanned by the image of S under the involution, is at least 2, then the number of edges in G is at least the Mantel-Turán bound, namely the number achieved by two disjoint cliques of …
The planar Turán number of $\{K_4,Θ_5\}$
本文研究禁用$K_4$和$\Theta_5$的平面图最大边数，证明其上界为$25/11(n-2)$，并通过构造说明该界对无穷多个图紧。
Random Turán Problems for Graphs with a Vertex Complete to One Part
本文研究了图的随机图论问题，证明了具有一个部分完全连接的顶点的 bipartite 图的紧界，同时提出了新的通用上界，改进了之前的结果。
Write $K^{(k)}_{n}$ for the complete $k$-graph on $n$ vertices. For $2 \leq k \leq g < r$ integers, let $π\left(n, K^{(k)}_{g}, K^{(k)}_r\right)$ be the maximum density of $K^{(k)}_{g}$ in $n$ vertex $K^{(k)}_{r}$-free $k$-graphs. The main contribution of this paper is the upper bound: $π\left(n, K^{(k)}_{g}, K^{(k)}_r\right) \leq \left(1 + O\left(n^{-1}\right) \right)\prod_{m=k}^{g} \left(1 - \frac{\binom{m-1}{k-1}}{\binom{r-1}{k-1}} \right).$ The graph case ($k=2$) is the first known generalized Turán question, investigated by Erdős. The $k=g$ case is the hypergraph Turán problem where the b…
For a positive integer $t$, let $F_t$ denote the graph of the $t\times t$ grid. Motivated by a 50-year-old conjecture of Erdős about Turán numbers of $r$-degenerate graphs, we prove that there exists a constant $C=C(t)$ such that $\mathrm{ex}(n,F_t)\leq Cn^{3/2}$. This bound is tight up to the value of $C$. One of the interesting ingredients of our proof is a novel way of using the tensor power trick.
Random Polynomial Graphs for Random Turán Problems
本文将 Bukh 和 Conlon 的随机多项式图方法推广到随机图 $G_{n,p}$，给出了不含平衡有根树幂 $\mathcal{T}^\ell$ 的子图最大边数及广义 Turán 数的有效下界。
A conditional lower bound for the Turán number of spheres
该研究提出了关于球体Turán数的条件下界，并通过归纳法给出了上界，推测上界可与下界匹配。
For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this paper, we define a variant of Turán number in hypergraphs, namely the cover Turán number, denoted as $\hat{ex}_R(n, G)$, as the maximum number of edges in the shadow graph of a Berge-$G$ free $R$-graph on $n$ vertices. We show a general…
Turán Number of Subdivisions of Multipartite Graphs
该文研究非二部非完全图的1-细分的Turán指数，给出了$K_{s,t}^+$（$4\leq s\leq t$）1-细分的Turán数上界，以及三类三部图1-细分构成图族的极值数上界。
Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle and path of length $k$, respectively. In this paper, we find sharp upper bounds for $ex(n,K_3,C_3^3)$ and the exact value for $ ex(n,K_3,P_3^3)$ and determine the graphs attaining these bounds.
Given graphs $F$ and $H$, the generalized rainbow Turán number $\text{ex}(n,F,\text{rainbow-}H)$ is the maximum number of copies of $F$ in an $n$-vertex graph with a proper edge-coloring that contains no rainbow copy of $H$. B. Janzer determined the order of magnitude of $\text{ex}(n,C_s,\text{rainbow-}C_t)$ for all $s\geq 4$ and $t\geq 3$, and a recent result of O. Janzer implied that $\text{ex}(n,C_3,\text{rainbow-}C_{2k})=O(n^{1+1/k})$. We prove the corresponding upper bound for the remaining cases, showing that $\text{ex}(n,C_3,\text{rainbow-}C_{2k+1})=O(n^{1+1/k})$. This matches the known…
The planar Turán number of a graph $H$, denoted by $ex_{_\mathcal{P}}(n,H)$, is the largest number of edges in a planar graph on $n $ vertices without containing $H$ as a subgraph. In this paper, we continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230]. We first obtain an improved lower bound for $ex_{_\mathcal{P}}(n,C_k)$ for all $k\ge 13$ and $n\ge 5(k-6+\lfloor{(k-1)}/2\rfloor)(k-1)/2$; the construction for each $k$ and $n$ provides a simpler counterexample to a conjecture of Ghosh, Győri, Martin, Paulos and Xiao [arxiv:2004.14094v1…
We study Turán and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {\em $\varepsilon$-balanced} if each color class contains at least an $\varepsilon$-proportion of its edges. Given a family $\mathcal{F}$ of edge-colored graphs, the Ramsey function $R(\varepsilon, \mathcal{F})$ is the smallest $n$ for which any $\varepsilon$-balanced $K_n$ must contain a copy of an $F\in\mathcal{F}$, and the Turán function $\mathrm{ex}(\varepsilon, n, \mathcal{F})$ is the maximum number of edges in an $n$-vertex $\varepsilon$-balanced graph which avoids all of $\mathcal{F}$. In thi…
Given a graph $F$, the planar Turán number of $F$, denoted $\text{ex}_{\mathcal{P}}(n, F)$, is the maximum number of edges in an $n$-vertex $F$-free planar graph. Such an extremal graph problem was initiated by Dowden while determining sharp upper bound for $\text{ex}_{\mathcal{P}}(n,C_4)$ and $\text{ex}_{\mathcal{P}}(n,C_5)$, where $C_4$ and $C_5$ are cycles of length four and five respectively. In this paper we determined an upper bound for $\text{ex}_{\mathcal{P}}(n,S_{2,2})$, $\text{ex}_{\mathcal{P}}(n,S_{2,3})$, $\text{ex}_{\mathcal{P}}(n,S_{2,4})$, $\text{ex}_{\mathcal{P}}(n,S_{2,5})$, $…
Upper Bounds on the Turán Density of Hypergraphs Associated with the Projective Plane over a Finite Field
本文利用 L^p 方法和射影平面的关联结构，得到了有限域射影平面相关超图的书状结构的 Turán 密度上界，并改进了已知结果。
An improved upper bound for planar Turán number of double star $S_{2,5}$
本文研究双星 $S_{2,5}$ 的平面 Turán 数，通过改进已有方法，对所有 $n\geq 1$ 证明了更优的尖锐上界 $\frac{19}{7}n-\frac{18}{7}$，并在 $n=12$ 时达到等号，从而改进了 Ghosh 等人的结果。
For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs $\mathcal{F}$, the relative Turán number $\mathrm{ex}(H,\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $H$. In this paper we give lower bounds on $\mathrm{ex}(H,\mathcal{F})$ for certain families of hypergraph cycles $\mathcal{F}$ such as Berge cycles and loose cycles. In particular, if $\mathcal{C}_\ell^3$ denotes the set of all $3$-uniform Berge $\ell$-cycles and $H$ is a 3-uniform hypergraph with maximum degree $Δ$, we prove \[\mathrm{ex}(H,\mathcal{C}_4^{3})\ge Δ^{-3/4-o(1)}e(H),\] …
The planar Turán number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph, which is obtained from $k$ triangles by sharing a common vertex. In this paper, we obtain sharp bounds of $ex_{\mathcal{P}}(n,H_k)$ and $ex_{\mathcal{P}}(n,K_1+P_{k+1})$ for $k\ge2$, which improves the results of Lan and Shi in Electron. J. Combin. 26 (2) (2019), \#P2.11.
In the so-called generalized Turán problems we study the largest number of copies of $H$ in an $n$-vertex $F$-free graph $G$. Here we introduce a variant, where $F$ is not forbidden, but we restrict how copies of $H$ and $F$ can be placed in $G$. More precisely, given an integer $n$ and graphs $H$ and $F$, what is the largest number of copies of $H$ in an $n$-vertex graph such that the vertex set of that copy does not contain and is not contained in the vertex set of a copy of $F$? We solve this problem for some instances, give bounds in other instances, and we use our results to determine the…
The classical Kővári-Sós-Turán theorem states that if $G$ is an $n$-vertex graph with no copy of $K_{s,t}$ as a subgraph, then the number of edges in $G$ is at most $O(n^{2-1/s})$. We prove that if one forbids $K_{s,t}$ as an induced/ subgraph, and also forbids any/ fixed graph $H$ as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a nontrivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a nontrivial upper bound on the number of…
Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete graph $K_r$: each vertex has been replaced with a set of size $k$. We determine the exact Turan number of $C^{2k}_3$ and the corresponding extremal hypergraph, thus confirming a conjecture of Frankl. Sidorenko has given an upper bound of $(r-2) / (r-1)$ for the Turán density of $C^{2k}_r$ for any $r$, and a construction establishing a matching lower bound w…
Ramsey and Turán numbers of sparse hypergraphs
本文定义超图的骨架退化性，证明其控制超图Turán数和Ramsey数，并给出线性Ramsey数及Turán数上下界，使用依赖随机选择技术。
Let $\mathrm{rex}(n, F)$ denote the maximum number of edges in an $n$-vertex graph that is regular and does not contain $F$ as a subgraph. We give lower bounds on $\mathrm{rex}(n, F)$, that are best possible up to a constant factor, when $F$ is one of $C_4$, $K_{2,t}$, $K_{3,3}$ or $K_{s,t}$ when $t>s!$.
Exact Turán numbers of two vertex-disjoint paths
本文确定了阶至少为4的两条顶点不相交奇数阶路径的精确Turán数，从而完全解决两条顶点不相交路径的Turán数问题，并证实了Yuan和Zhang在2021年提出猜想的$k=2$情形。
Turán problems for suspension of a balanced tree
本文基于Erdős-Sós猜想，确定了当$n\ge 4(4k)^6$时平衡树悬挂图的Turán数$\ex(n,\widehat{T})$的精确界，刻画了所有极图，并证明了一类分解族不含线性森林的图的Turán数。
An edge-colored graph $F$ is rainbow if each edge of $F$ has a unique color. The rainbow Turán number $ex^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Turán numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraëte in 2007. In this paper we focus on $ex^*(n,P_5)$. While several recent papers have investigated rainbow Turán numbers for $\ell$-edge paths $P_{\ell}$, exact results have only been obtained for $\ell < 5$, and $P_5$ represents one of the smallest cases left open i…
Let $H$ be a graph and $t\geq s\geq 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $λ(G) = O\left(n^{1-1/s}\right)$ where $λ(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.
Let $G$ be a connected graph of order $n$ with vertex set $V(G)$. A subset $S\subseteq V(G)$ is an $(a,b)$-dominating set if every vertex $v\in S$ is adjacent to at least $a$ vertices in $S$ and every $v\in V\setminus S$ is adjacent to at least $b$ vertices in $S$. The minimum cardinality of an $(a,b)$-dominating set of $G$ is the $(a,b)$-domination number of $G$, denoted by $γ_{a,b}(G)$. There are various results about upper bounds for $γ_{a,b}(G)$ when $G$ is regular or $a$ and $b$ are small numbers. In the first part of this paper, for a given graph $G$ with the minimum degree of $\max\{a,b…
Generalized Turán results for disjoint copies of degenerate graphs
本文扩展了广义Turán数研究，针对禁止图$(t+1)K_{a,b}$和$(t+1)C_{2k}$，建立了与已知结果同阶的精确匹配上下界。
For a fixed graph $F$, we would like to determine the maximum number of edges in a properly edge-colored graph on $n$ vertices which does not contain a {\emph rainbow copy} of $F$, that is, a copy of $F$ all of whose edges receive a different color. This maximum, denoted by $ex^*(n,F)$, is the {\emph rainbow Turán number} of $F$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte in 2007. We determine $ex^*(n,F)$ exactly when $F$ is a forest of stars, and give bounds on $ex^*(n,F)$ when $F$ is a path with $k$ edges, disproving a conjecture in Keevash et al.
The Johnson graph $J(n,m)$ has the $m$--subsets of $\{1,2,\ldots,n\}$ as vertices and two subsets are adjacent in the graph if they share $m-1$ elements. Shapozenko asked about the isoperimetric function $μ_{n,m}(k)$ of Johnson graphs, that is, the cardinality of the smallest boundary of sets with $k$ vertices in $J(n,m)$ for each $1\le k\le {n\choose m}$. We give an upper bound for $μ_{n,m}(k)$ and show that, for each given $k$ such that the solution to the Shadow Minimization Problem in the Boolean lattice is unique, and each sufficiently large $n$, the given upper bound is tight. We also sh…
New Bounds for Induced Turán Problems
本文研究了导出极值数的新界，利用退化度建立了其与标准极值数的非平凡关系，证明其由单个连通分量的2-核主导，并提出了可能作为反例的关联几何图。
Unbalanced Turán and spectral Turán problems with prescribed large maximum degree
本文研究指定最大度下无F图的不平衡Turán问题和谱Turán问题，针对不同类型的F给出极值图结论，建立边界与谱半径界的关联，丰富了Turán理论的相关结果。
Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Turán number of Berge-$\mathcal{F}$ is the maximum possible number of edges in an $r$-uniform hypergraph on $n$ vertices containing no Berge-$F$ as a subhypergraph (for every $F \in \mathcal{F}$) and is denoted by $ex_r(n,\mathcal{F})$. We determine the asymptotics for the Turán number of Berge-$K_{2,t}$ by showing $$ex_3(n,K_{2,t})=\frac{1}{6}(t-1)^{3/2} \cdot n^{3/2}(1+o(1))$$ for any given $t \ge 7$. We study the …
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