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28
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 ..."
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Cited by 39 (5 self)
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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.
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 ..."
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Cited by 26 (2 self)
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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.
Open problems of Paul Erdős in graph theory
 J. 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 a ..."
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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
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 ..."
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Cited by 15 (0 self)
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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.
The minimum size of 3graphs without a 4set spanning no or exactly three edges
 European J. Combin
"... Let Gi be the (unique) 3graph with 4 vertices and i edges. Razborov [On 3Hypergraphs with Forbidden 4Vertex Configurations, SIAM J. Discr. Math. 24 (2010), 946–963] determined asymptotically the minimum size of a 3graph on n vertices having neither G0 nor G3 as an induced subgraph. Here we obtai ..."
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Let Gi be the (unique) 3graph with 4 vertices and i edges. Razborov [On 3Hypergraphs with Forbidden 4Vertex Configurations, SIAM J. Discr. Math. 24 (2010), 946–963] determined asymptotically the minimum size of a 3graph on n vertices having neither G0 nor G3 as an induced subgraph. Here we obtain the corresponding stability result, determine the extremal function exactly, and describe all extremal hypergraphs for n ≥ n0. It follows that any sequence of almost extremal hypergraphs converges, which answers in the affirmative a question posed by Razborov. 1
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 ..."
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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 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 ..."
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Cited by 9 (2 self)
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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.
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 .... ..."
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Cited by 8 (0 self)
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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 ....
AN EXACT RESULT FOR HYPERGRAPHS AND UPPER BOUNDS FOR THE TURÁN DENSITY OF K r r+1
 SIAM J. DISCRETE MATH. VOL. 23, NO. 3, PP. 1324–1334
, 2009
"... We first answer a question of de Caen [Extremal Problems for Finite Sets, János Bolyai Math. Soc., Budapest, 1994, pp. 187–197]: given r ≥ 3, if G is an runiform hypergraph on n vertices such that every r + 1 vertices span 1 or r + 1 edges, then G = Kr n or Kr n−1, assuming that n>(p − 1)r, wher ..."
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Cited by 2 (0 self)
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We first answer a question of de Caen [Extremal Problems for Finite Sets, János Bolyai Math. Soc., Budapest, 1994, pp. 187–197]: given r ≥ 3, if G is an runiform hypergraph on n vertices such that every r + 1 vertices span 1 or r + 1 edges, then G = Kr n or Kr n−1, assuming that n>(p − 1)r, wherepis the smallest prime factor of r − 1. We then show that the Turán density π(K r r+1) ≤ 1 − 1/r − (1 − 1/rp−1)(r − 1) 2 /(2rp ( ( ) () r+p r+1 +)), for all even r ≥ 4, improving p−1 2 a wellknown bound 1 − 1 of de Caen [Ars Combin., 16 (1983), pp. 5–10] and Sidorenko [Vestnik r