A classical extremal, or Turán-type problem asks to determine ${\rm ex}(G, H)$, the largest number of edges in a subgraph of a graph $G$ which does not contain a subgraph isomorphic to $H$. Alon and Shikhelman introduced the so-called generalized extremal number ${\rm ex}(G,T,H)$, defined to be the maximum number of subgraphs isomorphic to $T$ in a subgraph of $G$ that contains no subgraphs isomorphic to $H$. In this paper we investigate the case when $G = Q_n$, the hypercube of dimension $n$, and $T$ and $H$ are smaller hypercubes or cycles.
In this paper we describe a number of extensions to Razborov's semidefinite flag algebra method. We will begin by showing how to apply the method to significantly improve the upper bounds of edge and vertex Turán density type results for hypercubes. We will then introduce an improvement to the method which can be applied in a more general setting, notably to 3-uniform hypergraphs, to get a new upper bound of 0.5615 for $π(K_4^3)$. For hypercubes we improve Thomason and Wagner's result on the upper bound of the edge Turán density of a 4-cycle free subcube to 0.60318 and Chung's result on forbid…
In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the…
Let $\mathcal{Q}_n$ be the $n$-dimensional hypercube: the graph with vertex set $\{0,1\}^n$ and edges between vertices that differ in exactly one coordinate. For $1\leq d\leq n$ and $F\subseteq \{0,1\}^d$ we say that $S\subseteq \{0,1\}^n$ is \emph{$F$-free} if every embedding $i:\{0,1\}^d\to \{0,1\}^n$ satisfies $i(F)\not\subseteq S$. We consider the question of how large $S\subseteq \{0,1\}^n$ can be if it is $F$-free. In particular we generalise the main prior result in this area, for $F=\{0,1\}^2$, due to E.A. Kostochka and prove a local stability result for the structure of near-extremal …
In this paper, we address a particular variation of the Turán problem for the hypercube. Alon, Krech and Szabó (2007) asked "In an n-dimensional hypercube, Qn, and for l < d < n, what is the size of a smallest set, S, of Q_l's so that every Q_d contains at least one member of S?" Likewise, they asked a similar Ramsey type question: "What is the largest number of colors that we can use to color the copies of Q_l in Q_n such that each Q_d contains a Q_l of each color?" We give upper and lower bounds for each of these questions and provide constructions of the set S above for some specific cases.
Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erdős-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of length $r$. We show that $s_{2m}(\mathbb{Z}_2^d) \leq C_m 2^{d/m} + O(1)$ when $d\rightarrow\infty$, and $s_{2m}(\mathbb{Z}_2^d) \geq 2^{d/m} + 2m-1$ when $d=km$. We use results on $s_r(G)$ to prove new bounds for the codegree Turán density of complete $r$-graphs.
We make some conjectures about extremal densities of daisy-free families, where a `daisy' is a certain hypergraph. These questions turn out to be related to some Turan problems in the hypercube, but they are also natural in their own right.
The b-chromatic number $b(G)$ of a graph $G$ is the maximum $k$ for which $G$ has a proper vertex coloring using $k$ colors such that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we mainly investigate on one of the open problems given in [P. Francis, S. Francis Raj, On b-coloring of powers of hypercubes, Discrete Appl. Math. 225 (2017) 74-86.]. As a consequence, we have obtained an upper bound for the b-chromatic number of some powers of hypercubes. This turns out to be an improvement of the already existing bound in [P. Francis,…
The present paper is concerned with the various algebraic structures supported by the set of Turán densities. We prove that the set of Turán densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a consequence we construct explicit irrational densities for any r >= 3. The proof relies on a technique recently developed by Pikhurko. We also show that the set of all Turán densities forms a graded ring, and from this we obtain a short proof of a theorem of Peng on jumps of hypergraphs. Finally, we prove that the set of Turán densities of families of r-graphs has po…
The Turán density $π(\cal F)$ of a family $\cal F$ of $r$-graphs is the limit as $n\to\infty$ of the maximum edge density of an $\cal F$-free $r$-graph on $n$ vertices. Erdos [Israel J. Math 2 (1964) 183--190] proved that no Turán density can lie in the open interval $(0,r!/r^r)$. Here we show that any other open subinterval of $[0,1]$ avoiding Turán densities has strictly smaller length. In particular, this implies a conjecture of Grosu [E-print arXiv:1403.4653v1, 2014].
Grosu [On the algebraic and topological structure of the set of Turán densities. \emph{J. Combin. Theory Ser. B} \textbf{118} (2016) 137--185] asked if there exist an integer $r\ge 3$ and a finite family of $r$-graphs whose Turán density, as a real number, has (algebraic) degree greater than~$r-1$. In this note we show that, for all integers $r\ge 3$ and $d$, there exists a finite family of $r$-graphs whose Turán density has degree at least~$d$, thus answering Grosu's question in a strong form.
The Turán density π(H) of a family H of k-graphs is the limit as n tends to infinity of the maximum edge density of an H-free k-graph on n vertices. Let I^k consist of all possible Turán densities and let F^k be the set of Turán densities of finite k-graph families. Here we prove that F^k contains every density obtained from an arbitrary finite construction by optimally blowing it up and using recursion inside the specified set of parts. As an application, we show that F^k contains an irrational number for each k\ge 3. Also, we show that I^k has cardinality of the continuum. In particular, I^k…
Upper Bounds on Turán Densities via Extremal Set Theory
Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
If $\mathcal{F}$ is a family of graphs then the Turán density of $\mathcal{F}$ is determined by the minimum chromatic number of the members of $\mathcal{F}$. The situation for Turán densities of 3-graphs is far more complex and still very unclear. Our aim in this paper is to present new exact Turán densities for individual and finite families of 3-graphs, in many cases we are also able to give corresponding stability results. As well as providing new examples of individual 3-graphs with Turán densities equal to 2/9,4/9,5/9 and 3/4 we also give examples of irrational Turán densities for finite …
School of Mathematics, Shandong University（山东大学数学学院）；Extremal Combinatorics and Probability Group, Institute for Basic Science（基础科学研究所极值组合与概率组）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）；Mathematics Institute and DIMAP, University of Warwick（华威大学数学研究所和DIMAP）
For integers $r \geq 3$ and $t \geq 2$, an $r$-uniform $t$-daisy $\mathcal{D}^t_r$ is a family of $\binom{2t}{t}$ $r$-element sets of the form $$\{S \cup T \ : T\subset U, \ |T|=t \}$$ for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S \cap U = \emptyset$. It was conjectured by Bollobás, Leader and Malvenuto (and independently Bukh) that the Turán densities of $t$-daisies satisfy $\lim\limits_{r \to \infty} π(\mathcal{D}_r^t) = 0$ for all $t \geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Turán densities of…
In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Turán's theorem using the connection. The connection of Lagrangians and Turán densities can be also used to prove the fundamental theorem of Erdös-Stone-Simonovits on Turán densities of graphs. Very recently, the study of Turán densities of non-uniform hypergraphs have been motivated by extremal poset problems. In this paper, we attempt to explore the applications of Lagrangian method in determining Turán densities of non-uniform hyperg…
For a fixed positive integer $n$ and an $r$-uniform hypergraph $H$, the Turán number $ex(n,H)$ is the maximum number of edges in an $H$-free $r$-uniform hypergraph on $n$ vertices, and the Lagrangian density of $H$ is defined as $π_λ(H)=\sup \{r! λ(G) : G \;\text{is an}\; H\text{-free} \;r\text{-uniform hypergraph}\}$, where $λ(G)$ is the Lagrangian of $G$. For an $r$-uniform hypergraph $H$ on $t$ vertices, it is clear that $π_λ(H)\ge r!λ{(K_{t-1}^r)}$. We say that an $r$-uniform hypergraph $H$ on $t$ vertices is perfect if $π_λ(H)= r!λ{(K_{t-1}^r)}$. Let $P_t=\{e_1, e_2, \dots, e_t\}$ be the …
A Jump in the Codegree Turán Densities of Long Tight Cycles
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Institute for Basic Science (IBS)（韩国基础科学研究所）；University of Illinois Chicago（伊利诺伊大学芝加哥分校）；Institute for Advanced Study（普林斯顿高等研究院）
Colored Turán densities and finite directed templates
School of Mathematics, Shandong University（山东大学数学学院）；Universidad Pública de Navarra（纳瓦拉公立大学）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极端组合学与概率组）
Given an $r$-graph $H$ on $h$ vertices, and a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$ in an $\mathcal{F}$-free $r$-graph on $n$ vertices. Then the \emph{Turán $H$-density} of $\mathcal{F}$ is the limit \[π_{H}(\mathcal{F})= \lim_{n\rightarrow \infty}\ex_{H}(n, \mathcal{F})/\binom{n}{h}. \] This generalises the notions of \emph{Turán density} (when $H$ is an $r$-edge), and \emph{inducibility} (when $\mathcal{F}$ is empty). Although problems of this kind have received some attention, very few results are k…
A real number $α\in [0, 1)$ is a jump for an integer $r\ge 2$ if there exists $c>0$ such that no number in $(α, α+ c)$ can be the Turán density of a family of $r$-uniform graphs. A classical result of Erd\H os and Stone \cite{ES} implies that that every number in $[0, 1)$ is a jump for $r=2$. Erd\H os \cite{E64} also showed that every number in $[0, r!/r^r)$ is a jump for $r\ge 3$ and asked whether every number in $[0, 1)$ is a jump for $r\ge 3$. Frankl and Rödl \cite{FR84} gave a negative answer by showing a sequence of non-jumps for every $r\ge 3$. After this, Erd\H os modified the question …
For fixed integers $r\ge 3, e\ge 3$, and $v\ge r+1$, let $f_r(n,v,e)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph in which the union of arbitrary $e$ distinct edges contains at least $v+1$ vertices. In 1973, Brown, Erdős and Sós proved that $f_r(n,er-(e-1)k,e)=Θ(n^k)$ and conjectured that the limit $\lim_{n\rightarrow\infty}\frac{f_3(n,e+2,e)}{n^2}$ always exists for all fixed integers $e\ge 3$. In 2020 Shangguan and Tamo conjectured that the limit $\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}$ always exists for all fixed integers $r>k\ge 2$ and $e\ge …
For fixed integers $r>k\ge 2,e\ge 3$, let $f_r(n,er-(e-1)k,e)$ be the maximum number of edges in an $r$-uniform hypergraph in which the union of any $e$ distinct edges contains at least $er-(e-1)k+1$ vertices. A classical result of Brown, Erdős and Sós in 1973 showed that $f_r(n,er-(e-1)k,e)=Θ(n^k).$ The degenerate Turán density is defined to be the limit (if it exists) $$π(r,k,e):=\lim_{n\rightarrow\infty}\frac{f_r(n,er-(e-1)k,e)}{n^k}.$$ Extending a recent result of Glock for the special case of $r=3,k=2,e=3$, we show that $$π(r,2,3):=\lim_{n\rightarrow\infty}\frac{f_r(n,3r-4,3)}{n^2}=\frac{…
The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $r-2$ new vertices. Keevash and Sidorenko~ have previously determined Turán densities of two families of hypergraph extensions. We determine the Turán numbers for these families, using classical stability techniques and new tools introduced in our earlier paper.
Let $H$ be a graph on $n$ vertices and let the blow-up graph $G[H]$ be defined as follows. We replace each vertex $v_i$ of $H$ by a cluster $A_i$ and connect some pairs of vertices of $A_i$ and $A_j$ if $(v_i,v_j)$ was an edge of the graph $H$. As usual, we define the edge density between $A_i$ and $A_j$ as $d(A_i,A_j)=\frac{e(A_i,A_j)}{|A_i||A_j|}.$ We study the following problem. Given densities $γ_{ij}$ for each edge $(i,j)\in E(H)$. Then one has to decide whether there exists a blow-up graph $G[H]$ with edge densities at least $γ_{ij}$ such that one cannot choose a vertex from each cluster…
In this paper we investigate density conditions for finding a complete $r$-uniform hypergraph $K_{r+1}^{(r)}$ on $r+1$ vertices in an $(r+1)$-partite $r$-uniform hypergraph $G$. First we prove an optimal condition in terms of the densities of the $(r+1)$ induced $r$-partite subgraphs of $G$. Second, we prove a version of this result where we assume that $r$-tuples of vertices in $G$ have their neighbours evenly distributed in $G$. Third, we also prove a counting result for the minimum number of copies of $K_{r+1}^{(r)}$ when $G$ satisfies our density bound, and present some open problems. A st…
