Turan Density Research Articles

The Turan density problem for hypergraphs

The Turan density problem for hypergraphs

Maximum Cliques of Hypergraphs and Polynomial Optimization

A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Bulo and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local (global) minimizers of a homogeneous polynomial function of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Turan densities of complete non-uniform hypergraphs.

A Note on Non-jumping Numbers for r-Uniform Hypergraphs

A real number \(\alpha \in [0,1)\) is a jump for an integer \(r\ge 2\) if there exists a constant \(c>0\) such that any number in \((\alpha , \alpha +c]\) cannot be the Turan density of a family of r-uniform graphs. Erdős and Stone showed that every number in [0,1) is a jump for \(r=2\). Erdős asked whether the same is true for \(r\ge 3\). Frankl and Rodl gave a negative answer by showing the existence of non-jumps for \(r\ge 3\). Recently, Baber and Talbot showed that every number in \([0.2299,0.2316)\bigcup [0.2871, \frac{8}{27})\) is a jump for \(r=3\) using Razborov’s flag algebra method. Pikhurko showed that the set of non-jumps for every \(r\ge 3\) has cardinality of the continuum. But, there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we show that \(1+\frac{r-1}{l^{r-1}}-\frac{r}{l^{r-2}}\) is a non-jump for \(r\ge 4\) and \(l\ge 3\) which generalizes some earlier results. We do not know whether the same result holds for \(r=3\). In fact, when \(r=3\) and \(l=3\), \(1+\frac{r-1}{l^{r-1}}-\frac{r}{l^{r-2}}={2 \over 9}\), and determining whether \({2 \over 9}\) is a jump or not for \(r=3\) is perhaps the most important unknown question regarding this subject. Erdős offered $500 for answering this question.

Connection Between Polynomial Optimization and Maximum Cliques of Non-Uniform Hypergraphs

In Motzkin and Straus (Canad. J. Math 498 17, 533540 1965) provided a connection between the order of a maximum clique in a graph G and the Lagrangian function of G. In Rota Bulo and Pelillo (Optim. Lett. 500 3, 287295 2009) extended the Motzkin-Straus result to r-uniform hypergraphs by establishing a one-to-one correspondence between local (global) minimizers of a family of homogeneous polynomial functions of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study similar optimization problems and obtain the connection to maximum cliques for {s, r}-hypergraphs and {p, s, r}-hypergraphs, which can be applied to obtain upper bounds on the Turan densities of the complete {s, r}-hypergraphs and {p, s, r}-hypergraphs.

Dense 3-uniform hypergraphs containing a large clique

An r-uniform graph G is dense if and only if every proper subgraph G′ of G satisfies λ(G′) < λ(G), where λ(G) is the Lagrangian of a hypergraph G. In 1980’s, Sidorenko showed that π(F), the Turan density of an r-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all dense F-hom-free r-uniform hypergraphs. This connection has been applied in the estimating Turan density of hypergraphs. When r = 2, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when r ≥ 3, it becomes much harder to estimate the Lagrangians of r-uniform hypergraphs and to characterize the structure of all dense r-uniform graphs. The main goal of this note is to give some sufficient conditions for 3-uniform graphs with given substructures to be dense. For example, if G is a 3-graph with vertex set [t]and m edges containing [t − 1](3), then G is dense if and only if $$m \geqslant \left( {\frac{{t - 1}}{3}} \right) + \left( {\frac{{t - 2}}{2}} \right) + 1$$ . We also give a sufficient condition on the number of edges for a 3-uniform hypergraph containing a large clique minus 1 or 2 edges to be dense.

On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs. Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques. This connection provided a new proof of Turan classical result on the Turan density of complete graphs. Since then, Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs. Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. They showed that if G is a 3-uniform graph with m edges containing a clique of order t − 1, then λ(G) = λ([t − 1](3)) provided \(\left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) \leqslant m \leqslant \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)\). They also conjectured: If G is an r-uniform graph with m edges not containing a clique of order t − 1, then λ(G) < λ([t − 1](r)) provided \(\left( {\begin{array}{*{20}{c}} {t - 1} \\ r \end{array}} \right) \leqslant m \leqslant \left( {\begin{array}{*{20}{c}} {t - 1} \\ r \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ {r - 1} \end{array}} \right)\). It has been shown that to verify this conjecture for 3-uniform graphs, it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with \(m = \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)\). Regarding this conjecture, we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t − 1](3)E(G)| = p, then λ(G) < λ([t− 1](3)) provided \(m = \left( {\begin{array}{*{20}{c}} {t - 1} \\ 3 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {t - 2} \\ 2 \end{array}} \right)\) and t ≥ 17p/2 + 11.

Hypergraphs with Zero Chromatic Threshold

Let F be an r-uniform hypergraph. The chromatic threshold of the family of F-free, r-uniform hypergraphs is the infimum of all non-negative reals c such that the subfamily of F-free, r-uniform hypergraphs H with minimum degree at least $$c \left( {\begin{array}{c}|V(H)| r-1\end{array}}\right) $$c|V(H)|r-1 has bounded chromatic number. The study of chromatic thresholds of various graphs has a long history, beginning with the early work of Erd?s and Simonovits. One interesting problem, first proposed by ?uczak and Thomass? and then solved by Allen, B?ttcher, Griffiths, Kohayakawa and Morris, is the characterization of graphs having zero chromatic threshold, in particular the fact that there are graphs with non-zero Turan density that have zero chromatic threshold. Here, we make progress on this problem for r-uniform hypergraphs, showing that a large class of hypergraphs have zero chromatic threshold in addition to exhibiting a family of constructions showing another large class of hypergraphs have non-zero chromatic threshold. Our construction is based on a particular product of the Bollobas---Erd?s graphs defined earlier by the authors.

Connection between a class of polynomial optimization problems and maximum cliques of non-uniform hypergraphs

In 1965, Motzkin and Straus provided a connection between the order of a maximum clique in a graph $$G$$G and a global solution of a quadratic optimization problem determined by $$G$$G which is called the Lagrangian function of $$G$$G. This connection and its extensions have been useful in both combinatorics and optimization. In 2009, Rota Bulo and Pelillo extended the Motzkin---Straus result to $$r$$r-uniform hypergraphs by establishing a one-to-one correspondence between local (global) minimizers of a family of homogeneous polynomial functions of degree $$r$$r (different from Lagrangian function) and the maximal (maximum) cliques of an $$r$$r-uniform hypergraph. In this paper, we study similar optimization problems related to non-uniform hypergraphs and obtain some extensions of their results to non-uniform hypergraphs. In particular, we provide a one-to-one correspondence between local (global) minimizers of a family of non-homogeneous polynomial functions and the maximal (maximum) cliques of $$\{1, r\}$${1,r}-hypergraphs. An application of a main result gives an upper bound on the Turan density of complete $$\{1, r\}$${1,r}-hypergraphs. The approach applied in the proof follows from the approach in Rota Bulo and Pelillo (2009).

Asymptotic Bounds for General Covering Designs

Given five positive integers and t where and a t- general covering design is a pair where X is a set of n elements (called points) and a multiset of k-subsets of X (called blocks) such that every p-subset of X intersects at least λ blocks of in at least t points. In this article we continue the work carried out by Etzion, Wei, and Zhang [Des. Codes Cryptogr. 5 (1995), 217–239] on the asymptotic covering density of general covering designs. We will present combinatorial constructions leading to new upper bounds on the asymptotic covering density of 4-(n, 4, 6, 1) general covering designs and 4- general covering designs with . The new bound on the asymptotic covering density of 4-(n, 4, 6, 1) general covering designs is equivalent to a new lower bound for the Turan density .

Some Motzkin–Straus type results for non-uniform hypergraphs

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in 1965. This connection and its extensions were applied in Turan problems of graphs and uniform hypergraphs. Very recently, the study of Turan densities of non-uniform hypergraphs has been motivated by extremal poset problems. Peng et al. showed a generalization of Motzkin---Straus result for $$\{1,2\}$${1,2}-graphs. In this paper, we attempt to explore the relationship between the Lagrangian of a non-uniform hypergraph and the order of its maximum cliques. We give a Motzkin---Straus type result for $$\{1,r\}$${1,r}-graphs. Moreover, we also give an extension of Motzkin---Straus theorem for $$\{1, r_2, \cdots , r_l\}$${1,r2,?,rl}-graphs.

ON possible Turán densities

The Turan density π(F) of a family F of k-graphs is the limit as n → ∞ of the maximum edge density of an F-free k-graph on n vertices. Let Π ∞ (k) consist of all possible Turan densities and let Π fin (k) ⊆ Π ∞ (k) be the set of Turan densities of finite k-graph families. Here we prove that Π fin (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 Π fin (k) contains an irrational number for each k ≥ 3. Also, we show that Π ∞ (k) has cardinality of the continuum. In particular, Π ∞ (k) ≠ Π fin (k) .

${\ell}$-Degree Turán Density

Let $H_n$ be a $k$-graph on $n$ vertices. For $0 \le \ell \ell >1$, the set of $\pi_{\ell}^k(\mathcal{F})$ is dense in the interval $[0,1)$. Hence, there is no “jump” for $\ell$-degree Turan density when $k>\ell >1$. We also give a lower bound on $\pi_{\ell}^k(\mathc...

On Lagrangians of r-uniform hypergraphs

A remarkable connection between the order of a maximum clique and the Lagrangian of a graph was established by Motzkin and Straus in Can J Math 17:533---540 (1965). This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It has been also applied in spectral graph theory. Estimating the Lagrangians of hypergraphs has been successfully applied in the course of studying the Turan densities of several hypergraphs as well. It is useful in practice if Motzkin---Straus type results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus' result to hypergraphs is false. We attempt to explore the relationship between the Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range. In this paper, we give some Motzkin---Straus type results for r-uniform hypergraphs. These results generalize and refine a result of Talbot in Comb Probab Comput 11:199---216 (2002) and a result in Peng and Zhao (Graphs Comb, 29:681---694, 2013).

Turán Densities of Some Hypergraphs Related to $K_{k+1}^{k}$

Let $B_i^{(k)}$ be the $k$-uniform hypergraph whose vertex set is of the form $S\cup T$, where $|S|=i$, $|T|=k-1$, and $S\cap T=\emptyset$, and whose edges are the $k$-subsets of $S\cup T$ that contain either $S$ or $T$. We derive upper and lower bounds for the Turan density of $B_i^{(k)}$ that are close to each other as $k\to\infty$. We also obtain asymptotically tight bounds for the Turan density of several other infinite families of hypergraphs. The constructions that imply the lower bounds are derived from elementary number theory by probabilistic arguments, and the upper bounds follow from some results of de Caen, Sidorenko, and Keevash.

On Jumping Densities of Hypergraphs

A number $${\alpha\in [0, 1)}$$ is a jump for an integer r ≥ 2 if there exists a constant c > 0 such that for any family $${{\mathcal F}}$$of r-uniform graphs, if the Turan density of $${{\mathcal F}}$$is greater than α, then the Turan density of $${{\mathcal F}}$$is at least α + c. A fundamental result in extremal graph theory due to Erdős and Stone implies that every number in [0, 1) is a jump for r = 2. Erdős also showed that every number in [0, r!/r r ) is a jump for r ≥ 3. However, not every number in [0, 1) is a jump for r ≥ 3. In fact, Frankl and Rodl showed the existence of non-jumps for r ≥ 3. By a similar approach, more non-jumps were found for some r ≥ 3 recently. But there are still a lot of unknowns regarding jumps for hypergraphs. In this note, we show that if $${c\cdot{\frac{r!}{r^r}}}$$is a non-jump for r ≥ 3, then for every p ≥ r, $${c\cdot{\frac{p!}{p^p}}}$$is a non-jump for p.

A note on the Structure of Turán Densities of Hypergraphs

Let r ≥ 2 be an integer. A real number α ∈ [0, 1) is a jump for r if there exists c > 0 such that no number in (α, α + c) can be the Turan density of a family of r-uniform graphs. A result of Erdős and Stone implies that every α ∈ [0, 1) is a jump for r = 2. Erdős asked whether the same is true for r ≥ 3. Frankl and Rodl gave a negative answer by showing an infinite sequence of non-jumps for every r ≥ 3. However, there are still a lot of open questions on determining whether or not a number is a jump for r ≥ 3. In this paper, we first find an infinite sequence of non-jumps for r = 4, then extend one of them to every r ≥ 4. Our approach is based on the techniques developed by Frankl and Rodl.

Supersaturation For Ramsey-Turán Problems

For an l-graph $${\user1{\mathcal{G}}}$$, the Turan number $${\text{ex}}{\left( {n,{\user1{\mathcal{G}}}} \right)}$$ is the maximum number of edges in an n-vertex l-graph $${\user1{\mathcal{H}}}$$ containing no copy of $${\user1{\mathcal{G}}}$$. The limit $$\pi {\left( {\user1{\mathcal{G}}} \right)} = \lim _{{n \to \infty }} {\text{ex}}{\left( {n,{\user1{\mathcal{G}}}} \right)}/{\left( {^{n}_{l} } \right)}$$ is known to exist [8]. The Ramsey–Turan density $$p{\left( {\user1{\mathcal{G}}} \right)}$$ is defined similarly to $$\pi {\left( {\user1{\mathcal{G}}} \right)}$$ except that we restrict to only those $${\user1{\mathcal{H}}}$$ with independence number o(n). A result of Erdős and Sos [3] states that $$\pi {\left( {\user1{\mathcal{G}}} \right)} = p{\left( {\user1{\mathcal{G}}} \right)}$$ as long as for every edge E of $${\user1{\mathcal{G}}}$$ there is another edge E′of $${\user1{\mathcal{G}}}$$ for which |E∩E′|≥2. Therefore a natural question is whether there exists $${\user1{\mathcal{G}}}$$ for which $$p{\left( {\user1{\mathcal{G}}} \right)} 2.We prove the existence of a phenomenon similar to supersaturation for Turan problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs $${\user1{\mathcal{G}}}$$ for which $$0 < \ifmmode\expandafter\tilde\else\expandafter\~\fi{p}{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}$$.We also prove that the 3-graph $${\user1{\mathcal{G}}}$$ with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies $$0 < p{\left( {\user1{\mathcal{G}}} \right)} < \pi {\left( {\user1{\mathcal{G}}} \right)}$$. The existence of a hypergraph $${\user1{\mathcal{H}}}$$ satisfying $$0 < p{\left( {\user1{\mathcal{H}}} \right)} < \pi {\left( {\user1{\mathcal{H}}} \right)}$$ was conjectured by Erdős and Sos [3], proved by Frankl and Rodl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs.

On A Hypergraph Turán Problem Of Frankl

Let $$C^{{{\left( {2k} \right)}}}_{r}$$ be the 2k-uniform hypergraph obtained by letting P1, . . .,Pr be pairwise disjoint sets of size k and taking as edges all sets Pi∪Pj with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph Kr: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain $$C^{{{\left( {2k} \right)}}}_{3}$$ can be obtained by partitioning V = V1 ∪V2 and taking as edges all sets of size 2k that intersect each of V1 and V2 in an odd number of elements. Let $${\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n}$$ denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no $$C^{{{\left( {2k} \right)}}}_{3}$$ has at most as many edges as $${\user1{\mathcal{B}}}^{{{\left( {2k} \right)}}}_{n}$$.Sidorenko has given an upper bound of $$\frac{{r - 2}}{{r - 1}}$$ for the Tur´an density of $$C^{{{\left( {2k} \right)}}}_{r}$$ for any r, and a construction establishing a matching lower bound when r is of the form 2p+1. In this paper we also show that when r=2p+1, any $$C^{{{\left( 4 \right)}}}_{r}$$-free hypergraph of density $$\frac{{r - 2}}{{r - 1}} - o{\left( 1 \right)}$$ looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turan density of $$C^{{{\left( 4 \right)}}}_{r}$$ to $$\frac{{r - 2}}{{r - 1}} - c{\left( r \right)}$$, where c(r) is a constant depending only on r.The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.

The Turan Density of Triple Systems Is Not Principal

The Erdős–Stone–Simonovits Theorem implies that the Turán density of a family of graphs is the minimum of the Turán densities of the individual graphs from the family. It was conjectured by Mubayi and Rödl ( J. Combin. Theory Ser. A, submitted) that this is not necessarily true for hypergraphs, in particular for triple systems. We give an example, which shows that their conjecture is true.