Hypergraph Turán problems

Keevash, Peter

doi:10.1017/cbo9781139004114.004

Cited by 202 publications

(211 citation statements)

References 128 publications

Supporting

Mentioning

200

Contrasting

Unclassified

Order By: Relevance

“…It is well-known (see, e.g., [20]) that π H = lim n→∞ ex(n, H)/ n k exists, with π H ∈ [0, 1), and that for graphs (i.e., k = 2) we have π H = 1 − 1/(χ(H) − 1), where χ(H) is the chromatic number of H. Theorem 4. Let G ⊆ H be k-graphs with e G 1.…”

Section: Main Examplementioning

confidence: 99%

The Lower Tail: Poisson Approximation Revisited

Janson¹,

Warnke²

2017

Trends in Mathematics

View full text Add to dashboard Cite

The well-known "Janson's inequality" gives Poisson-like upper bounds for the lower tail probability P(X (1 − ε)EX) when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation-based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ε = 1) of Janson, Luczak and Ruciński.

show abstract

Section: Main Examplementioning

confidence: 99%

The Lower Tail: Poisson Approximation Revisited

Janson¹,

Warnke²

2017

Trends in Mathematics

View full text Add to dashboard Cite

show abstract

“…In particular, for the first step we use the following Turán-type result for hypergraphs, proved by de Caen [dC83] (see also [Kee11]). We note that a slightly weaker result can be proved by a simple probabilistic argument.…”

Section: Improved Bound Onmentioning

confidence: 99%

On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

Keller¹,

Smorodinsky²,

Tardos³

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

Let HD d (p, q) denote the minimal size of a transversal that can always be guaranteed for a family of compact convex sets in R d which satisfy the (p, q)-property (p ≥ q ≥ d + 1). In a celebrated proof of the Hadwiger-Debrunner conjecture, Alon and Kleitman proved thatThis paper has two parts. In the first part we present several improved bounds on HD d (p, q). In particular, we obtain the first near tight estimate of HD d (p, q) for an extended range of values of (p, q) since the 1957 Hadwiger-Debrunner theorem.In the second part we prove a (p, 2)-theorem for families in R 2 with union complexity below a specific quadratic bound. Based on this, we introduce a polynomial time constant factor approximation algorithm for MAX-CLIQUE of intersection graphs of convex sets satisfying this property. It is not likely that our constant factor approximation can be improved to a PTAS as MAX-CLIQUE for intersection graphs of fat ellipses is known to be APX-HARD and fat ellipses have sub-quadratic union complexity.

show abstract

“…where, given a complete t-uniform hypergraph in p vertices, denoted by K t p , and a positive integer n, ex(n, K t p ) denotes the maximum number of edges in a t-uniform hypergraph on n vertices which does not contain K t p as a subgraph. For a survey on hypergraph Turán problems, known results and definitions, the reader is referred to [28]. The problem of determining ex(n, K t p ) can thus be seen as the "dual" of the problem of determining T (n, t, p) for which ex(n, K t p ) + T (n, t, p) = n t and π(K t p ) + t(p, t) = 1 hold.…”

Section: (N 4 P) Turán Systemsmentioning

confidence: 99%

“…Proof. We will build ({n, v}, 4, 6) 5-uniform stretched covers for which (28) applies. The choice of the values of v will be similar in spirit to the one of Theorem 5.5.…”

Section: The Case K =mentioning

confidence: 99%

Asymptotic Bounds for General Covering Designs

Montecalvo

2014

J. Combin. Designs

View full text Add to dashboard Cite

Given five positive integers n, p, k, λ, and t where n ≥ k ≥ t and n ≥ p ≥ t, a t-(n, k, p, λ) general covering design is a pair (X, B) where X is a set of n elements (called points) and B a multiset of k-subsets of X (called blocks) such that every p-subset of X intersects at least λ blocks of B 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-(n, 5, p, 1) general covering designs with 5 ≤ p ≤ 6. 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 Turán density π (K 4 6 ).

show abstract

Hypergraph Turán problems

Cited by 202 publications

References 128 publications

The Lower Tail: Poisson Approximation Revisited

The Lower Tail: Poisson Approximation Revisited

On Max-Clique for intersection graphs of sets and the Hadwiger-Debrunner numbers

Asymptotic Bounds for General Covering Designs

Contact Info

Product

Resources

About