Results 1  10
of
16
New Constructions for Covering Designs
 J. Combin. Designs
, 1995
"... A (v,k,t) covering design, or covering, is a family of ksubsets, called blocks, chosen from a vset, such that each tsubset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by C(v,k,t). This paper gives thre ..."
Abstract

Cited by 20 (2 self)
 Add to MetaCart
A (v,k,t) covering design, or covering, is a family of ksubsets, called blocks, chosen from a vset, such that each tsubset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by C(v,k,t). This paper gives three new methods for constructing good coverings: a greedy algorithm similar to Conway and Sloane’s algorithm for lexicographic codes [6], and two methods that synthesize new coverings from preexisting ones. Using these new methods, together with results in the literature, we build tables of upper bounds on C(v,k,t) for v ≤ 32, k ≤ 16, and t ≤ 8. 1
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
Abstract

Cited by 13 (0 self)
 Add to MetaCart
Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Open problems of Paul Erdős in Graph Theory
 JOURNAL OF GRAPH THEORY
, 1997
"... The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory an ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory and beyond (e.g., in number theory, probability, geometry, algorithms and complexity theory). Solutions or partial solutions to Erdős problems usually lead to further questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists. Through the problems, the legacy of Paul Erdős continues (particularly if solving one of these problems results in creating three new problems, for example.) There is a huge literature of almost 1500 papers written by Erdős and his (more than 460) collaborators. Paul wrote many problem papers, some of which appeared in various (really hardtofind) proceedings. Here is an attempt to collect and organize these problems in the area of graph theory. The list here is by no means complete or exhaustive. Our goal is to state the problems, locate the sources, and provide the references related to these problems. We will include the earliest and latest known references without covering the entire history of the problems because of space limitations. (The most uptodate list of Erdős’ papers can be found in [65]; an electronic file is maintained by Jerry Grossman at
On 3hypergraphs with forbidden 4vertex configurations
, 2010
"... Every 3graph in which no four vertices are independent and no four vertices span precisely three edges must have edge density ≥ 4/9(1 − o(1)). This bound is tight. The proof is a rather elaborate application of CauchySchwarz type arguments presented in the framework of flag algebras. We include fu ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
Every 3graph in which no four vertices are independent and no four vertices span precisely three edges must have edge density ≥ 4/9(1 − o(1)). This bound is tight. The proof is a rather elaborate application of CauchySchwarz type arguments presented in the framework of flag algebras. We include further demonstrations of this method by reproving a few known tight results about hypergraph Turán densities and significantly improving numerical bounds for several problems for which the exact value is not known yet.
An upper bound for the Turan number t_3(n, 4)
, 2000
"... Let t r (n, r +1) denote the smallest integer m such that every runiform hypergraph on n vertices with m+ 1 edges must contain a complete graph on r + 1 vertices. In this paper, we prove that lim n## t 3 (n, 4) # n 3 # # 3+ # 17 12 = .593592 .... ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
Let t r (n, r +1) denote the smallest integer m such that every runiform hypergraph on n vertices with m+ 1 edges must contain a complete graph on r + 1 vertices. In this paper, we prove that lim n## t 3 (n, 4) # n 3 # # 3+ # 17 12 = .593592 ....
On the FonderFlaass Interpretation of Extremal Examples for Turán’s (3,4)problem
, 2010
"... In 1941, Turán conjectured that the edge density of any 3graph without independent sets on 4 vertices (Turán (3, 4)graph) is ≥ 4/9(1− o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many other examples of this density. FonderFlaass (1988) prese ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In 1941, Turán conjectured that the edge density of any 3graph without independent sets on 4 vertices (Turán (3, 4)graph) is ≥ 4/9(1− o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many other examples of this density. FonderFlaass (1988) presented a general construction that converts an arbitrary ⃗ C4free orgraph Γ into a Turán (3, 4)graph. He observed that all TuránBrownKostochka examples result from his construction, and proved the bound ≥ 3/7(1 − o(1)) on the edge density of any Turán (3, 4)graph obtainable in this way. In this paper we establish the optimal bound 4/9(1 − o(1)) on the edge density of any Turán (3, 4)graph resulting from the FonderFlaass construction under any of the following assumptions on the undirected graph G underlying the orgraph Γ: • G is complete multipartite; • the edge density of G is ≥ (2/3 − ɛ) for some absolute constant ɛ> 0. We are also able to improve FonderFlaass’s bound to 7/16(1 − o(1)) without any extra assumptions on Γ.
On Hypergraphs of Girth Five
 ELECTRONIC JOURNAL OF COMBINATORICS
, 2003
"... In this paper, we study runiform hypergraphs without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for r = 3, we show that if has n vertices and a maximum number of edges, then 1 ). This also asymptotically determines the gene ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
In this paper, we study runiform hypergraphs without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for r = 3, we show that if has n vertices and a maximum number of edges, then 1 ). This also asymptotically determines the generalized Turan number T 3 (n, 8, 4). Some results are based on our bounds for the maximum size of Sidontype sets in n .
On an Extremal Hypergraph Problem of Brown, Erdős and Sós
 COMBINATORICA
, 2005
"... Let f r (n, v, e) denote the maximum number of edges in an runiform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemeredi and of Erdős, Frankl and Rödl, we partially resolve a problem raised by Brown, Erdős and Sós in 19 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Let f r (n, v, e) denote the maximum number of edges in an runiform hypergraph on n vertices, which does not contain e edges spanned by v vertices. Extending previous results of Ruzsa and Szemeredi and of Erdős, Frankl and Rödl, we partially resolve a problem raised by Brown, Erdős and Sós in 1973, by showing that for any fixed 2 k < r, we have ).
An upper bound for the Turan number
"... Let t r (n, r +1) denote the smallest integer m such that every runiform hypergraph on n vertices with m+ 1 edges must contain a complete graph on r + 1 vertices. In this paper, we prove that lim n## t 3 (n, 4) # n 3 # # 3+ # 17 12 = .593592 .... 1 Introduction For an runiform hypergraph H ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Let t r (n, r +1) denote the smallest integer m such that every runiform hypergraph on n vertices with m+ 1 edges must contain a complete graph on r + 1 vertices. In this paper, we prove that lim n## t 3 (n, 4) # n 3 # # 3+ # 17 12 = .593592 .... 1 Introduction For an runiform hypergraph H (or rgraph, for short), we denote by t r (n, H) the smallest integer m such that every rgraph on n vertices with m + 1 edges must contain H as a subgraph. When H is a complete graph on k vertices, we write t r (n, k)=t r (n, H). In 1941, Turan [10] determined the Turan number t 2 (n, k) for 2graphs and he asked the problem of determining the limit lim n## t r (n, k) # n r # , for 2 <r<k. For this problem, Erdos o#ered $1000 in honor of Paul Turan, (see [1] and [10]). Since 1941, the above problem has remained open, even for the first nontrivial case of r =3andk = 4. The exact value for Turan number t 3 (n, 4) is conjectured [10] as follows: # Journal of Combinatorial Theory (A),...