A (v,k,t) covering design, or covering, is a family of k-subsets, called blocks, chosen from a v-set, such that each t-subset 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

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 hard-to-find) 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 up-to-date list of Erdős’ papers can be found in [65]; an electronic file is maintained by Jerry Grossman at

Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as H|S = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VC-dimension) of H is the size of the largest subset S for which H|S has 2 |S| edges. Hypergraphs of small VC-dimension 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.

Let t r (n, r +1) denote the smallest integer m such that every r-uniform 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 ....

In this paper, we study r-uniform 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 Sidon-type sets in n .

Abstract not found

Let t r (n, r +1) denote the smallest integer m such that every r-uniform 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 r-uniform hypergraph H (or r-graph, for short), we denote by t r (n, H) the smallest integer m such that every r-graph 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 2-graphs 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 non-trivial 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),...

In 1941, Turán conjectured that the edge density of any 3-graph 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. Fon-der-Flaass (1988) presented a general construction that converts an arbitrary ⃗ C4-free orgraph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka 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 Fon-der-Flaass 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 Fon-der-Flaass’s bound to 7/16(1 − o(1)) without any extra assumptions on Γ.

r( n, H) the smallest integer m such that every r-graph 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 [?] determined the Turan number t 2( n, k) for 2-graphs 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 [2] and [7]). Since 1941, the above problem has remained open, even for the first non-trivial case of r =3andk = 4. For small values of n, the conj

For Turán’s (3, 4)-conjecture, in the case of n = 3k + 1 vertices, 1 2 6k−1 nonisomorphic hypergraphs are constructed that attain the conjecture. In the case of n = 3k + 2 vertices, 6 k−1 non-isomorphic hypergraphs are constructed that attain the conjecture. 1