Dense 3-uniform hypergraphs containing a large clique

An r-uniform graph C 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 Turán density of an γ-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all dense F-hom-free γ-uniform hypergraphs. This connection has been applied in the estimating Turán density of hypergraphs. When γ=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 γ-uniform hypergraphs and to characterize the structure of all dense γ-uniform graphs. The main goal of this note is to give some sufficient conditions for3-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≥{t-2 3)+(t-2 2)+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.     

Embeddings and Ramsey numbers of sparse <Emphasis Type="Italic">κ</Emphasis>-uniform hypergraphs

Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6,23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to κ-uniform hypergraphs for any integer κ ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for κ-uniform hypergraphs of bounded maximum degree into suitable κ-uniform ‘quasi-random’ hypergraphs.     

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 Turán density of \({{\mathcal F}}\) is greater than α, then the Turán density of \({{\mathcal F}}\) is at least α + c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0, 1) is a jump for r = 2. Erd?s 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 Rödl 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.     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

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 Turán density of a family of r-uniform graphs. A result of Erd?s and Stone implies that every α ∈ [0, 1) is a jump for r = 2. Erd?s asked whether the same is true for r ≥ 3. Frankl and Rödl 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 Rödl.     

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 Buló 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 Turán densities of complete non-uniform hypergraphs.     

Generalisations of Hypomorphisms and Reconstruction of Hypergraphs

We define certain generalisations of hypergraph hypomorphisms, which we call k-morphisms, \((k,n-k)\)-hypomorphisms, partial \((k,n-k)\)-hypomorphisms. They are special bijections between collections of k-subsets of vertex sets of hypergraphs. We show that these mappings lead to alternative representations of the automorphism groups of r-uniform hypergraphs and vertex stabilisers of graphs. We also use them to show that almost every r-uniform hypergraph is reconstructible and \((k,n-k)\)-reconstructible. As a consequence we also obtain the result that almost every r-uniform hypergraph is asymmetric.     

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533–540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false. Frankl and Füredi conjectured that the r-uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of \({\mathbb N}^{(r)}\) has the largest Lagrangian of all r-uniform hypergraphs with m edges. For \(r=2\), Motzkin and Straus’ theorem confirms this conjecture. For \(r=3\), it is shown by Talbot that this conjecture is true when m is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for 3-uniform hypergraphs. As an application of this connection, we confirm that Frankl and Füredi’s conjecture holds for bigger ranges of m when \(r=3\). We also obtain two weaker versions of Turán type theorem for left-compressed 3-uniform hypergraphs.     

New Families of <Emphasis Type="Italic">n</Emphasis>-Clusters Verifying the Erdős–Faber–Lovász Conjecture

Erd?s, Faber and Lovász conjectured in 1972 that the vertices of a linear hypergraph with n edges, each of size n, can be strongly colored with n colors. It was shown by Romero and Sánchez-Arroyo that an equivalent conjecture is obtained when linear hypergraphs are replaced by n-clusters. In this paper we describe new families of EFL-compliant n-clusters; that is, those for which the conjecture holds. Moreover, we describe ways to extend some n-clusters to larger ones preserving EFL-compliance. Also, our approach allowed us to provide a new upper bound for the chromatic number of n-clusters.     

Chromatic number and subtrees of graphs

Let G and H be two graphs. We say that G induces H if G has an induced subgraph isomorphic to H: A. Gyárfás and D. Sumner, independently, conjectured that, for every tree T. there exists a function f _T ; called binding function, depending only on T with the property that every graph G with chromatic number f _T (ω(G)) induces T. A. Gyárfás, E. Szemerédi and Z. Tuza confirmed the conjecture for all trees of radius two on triangle-free graphs, and H. Kierstead and S. Penrice generalized the approach and the conclusion of A. Gyárfás et al. onto general graphs. A. Scott proved an interesting topological version of this conjecture asserting that for every integer k and every tree T of radius r, every graph G with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(14 ^r-1(r - 1)!) times. We extend the approach of A. Gyárfás and present a binding function for trees obtained by identifying one end of a path and the center of a star. We also improve A. Scott's upper bound by modifying his subtree structure and partition technique, and show that for every integer k and every tree T of radius r, every graph with ω(G) ? k and sufficient large chromatic number induces a subdivision of T of which each edge is subdivided at most O(6 ^r?2) times.     

On a generalization of the Hadwiger–Nelson problem

For a field F and a quadratic form Q defined on an n-dimensional vector space V over F, let QG _Q , called the quadratic graph associated to Q, be the graph with the vertex set V where vertices u,w ∈ V form an edge if and only if Q(v ? w) = 1. Quadratic graphs can be viewed as natural generalizations of the unit-distance graph featuring in the famous Hadwiger–Nelson problem. In the present paper, we will prove that for a local field F of characteristic zero, the Borel chromatic number of QG _Q is infinite if and only if Q represents zero non-trivially over F. The proof employs a recent spectral bound for the Borel chromatic number of Cayley graphs, combined with an analysis of certain oscillatory integrals over local fields. As an application, we will also answer a variant of question 525 proposed in the 22nd British Combinatorics Conference 2009 [6].     

The independent neighborhoods process

A triangle T^(r) in an r-uniform hypergraph is a set of r+1 edges such that r of them share a common (r-1)-set of vertices and the last edge contains the remaining vertex from each of the first r edges. Our main result is that the random greedy triangle-free process on n points terminates in an r-uniform hypergraph with independence number O((n log n)^1/r). As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number r(T^(r),K_s^(r)) has order of magnitude s^r/ log s. This answers questions posed in [4, 10] and generalizes the celebrated results of Ajtai–Komlós–Szemerédi [1] and Kim [9] to hypergraphs.     

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 Bulò 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 Turán densities of the complete {s, r}-hypergraphs and {p, s, r}-hypergraphs.     

Independence numbers of random subgraphs of distance graphs

We consider the distance graph G(n, r, s), whose vertices can be identified with r-element subsets of the set {1, 2,..., n}, two arbitrary vertices being joined by an edge if and only if the cardinality of the intersection of the corresponding subsets is s. For s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s–Ko–Rado problem and also play an important role in combinatorial geometry and coding theory. We study some properties of random subgraphs of G(n, r, s) in the Erd?s–Rényi model, in which every edge occurs in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, r, s) for the case of constant r and s. The independence number of a random subgraph is Θ(log₂n) times as large as that of the graph G(n, r, s) itself for r ≤ 2s + 1, while for r > 2s + 1 one has asymptotic stability: the two independence numbers asymptotically coincide.     

On non-strong jumping numbers and density structures of hypergraphs

Estimating Turán densities of hypergraphs is believed to be one of the most challenging problems in extremal set theory. The concept of ‘jump’ concerns the distribution of Turán densities. A number α∈[0,1) is a jump for r-uniform graphs if there exists a constant c>0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Turán density of F is at least α+c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0,1) is a jump for graphs. Erd?s also showed that every number in [0,r!/r^r) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in [r!/r^r,1) for r-uniform hypergraphs. But there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we propose a new but related concept-strong-jump and describe several sequences of non-strong-jumps. It might help us to understand the distribution of Turán densities for hypergraphs better by finding more non-strong-jumps.     

Learning a hidden uniform hypergraph

Motivated by applications in genome sequencing, Grebinski and Kucherov (Discret Appl Math 88:147–165, 1998) studied the graph learning problem which is to identify a hidden graph drawn from a given class of graphs with vertex set \(\{1,2,\ldots ,n\}\) by edge-detecting queries. Each query tells whether a set of vertices induces any edge of the hidden graph or not. For the class of all hypergraphs whose edges have size at most r, Chodoriwsky and Moura (Theor Comput Sci 592:1–8, 2015) provided an adaptive algorithm that learns the class in \(O(m^r\log n)\) queries if the hidden graph has m edges. In this paper, we provide an adaptive algorithm that learns the class of all r-uniform hypergraphs in \(mr\log n+(6e)^rm^{\frac{r+1}{2}}\) queries if the hidden graph has m edges.     

Ordering uniform supertrees by their spectral radii

A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.     

Improved Upper Bound for Generalized Acyclic Chromatic Number of Graphs

A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|, r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree Δ ≥ 7 and with girth at least (r ? 1)Δ is at most (4r ? 3)Δ.     

On countable unions of nonmeager sets in hereditarily Lindelöf spaces

It is well known that any Vitali set on the real line ? does not possess the Baire property. The same is valid for finite unions of Vitali sets. What can be said about infinite unions of Vitali sets? Let S be a Vitali set, S _r be the image of S under the translation of ? by a rational number r and F = {S _r : r is rational}. We prove that for each non-empty proper subfamily F′ of F the union ∪F′ does not possess the Baire property. We say that a subset A of ? possesses Vitali property if there exist a non-empty open set O and a meager set M such that A ? O \ M. Then we characterize those non-empty proper subfamilies F′ of F which unions ∪F′ possess the Vitali property.     

Universal graphs with a forbidden subgraph: Block path solidity

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Füredi and Komjáth, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.