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 $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m \geq 2t \geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $\mathbb{Z}/m\mathbb{Z}$ such that for any $x \in \mathbb{Z}/m\mathbb{Z}$ and any $2t$-element subset $X$ of $\mathbb{Z}/m\mathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of the extremal Hilbert cube problem considered by Gunderson and Rödl and its generalization studied by Cilleruelo and Tesoro.
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.
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；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）；Hefei National Laboratory, University of Science and Technology of China（中国科学技术大学合肥国家实验室）
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）
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 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…
For integers $q\ge p\ge r\ge2$, we say that an $r$-uniform hypergraph $H$ has property $(q,p)$, if for any $q$-vertex subset $Q$ of $V(H)$, there exists a $p$-vertex subset $P$ of $Q$ spanning a clique in $H$. Let $T_{r}(n,q,p)=\min\{ e(H): H\subset \binom{[n]}{r}, H \text{~has property~} (q,p)\}$. The local Turán density about property $(q,p)$ in $r$-uniform hypergraphs is defined as $t_{r}(q,p)=\lim_{n\to \infty}T_{r}(n,q,p)/\binom{n}{r}$. Frankl, Huang and Rödl [J. Comb. Theory, Ser. A, 177 (2021)] showed that $\lim_{p\to\infty}t_{r}(ap+1,p+1)=\frac{1}{a^{r-1}}$ for positive integer $a$ and…
In this paper, we consider the Turán problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Turán densities of some small $\{1,3\}$-hypergraphs.
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…
Leipzig University（莱比锡大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学研究所）；Masaryk University（马萨里克大学）；Institute for Basic Science (IBS)（基础科学研究院）
In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers $\mathbb N$ as limiting values of $q$-series as $q\to ζ$ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of $\mathbb N$ by analogous structures in the integer partitions $\mathcal P$. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of…
