What we know and what we do not know about Turán numbers

Abstract

Abstract. The numbers which are traditionally named in honor of Paul Turfin were introduced by him as a generalization of a problem he solved in 1941. The general problem of Tur~in having an extremely simple formulation but being extremely hard to solve, has become one of the most fascinating extremal problems in combinatorics. We describe the present situation and list conjectures which are not so hopeless. The Definition and Equivalent FormulationsA system of r-element subsets (blocks) of an n-element set Xn is called a Turin (n, k, r)-system if every k-element subset of X, contains at least one of the blocks. The Turrn number T(n, k, r) is the minimum size of such a system. The problem of determining T(n, k, r) was posed by Paul Turin [57].The Tur~m numbers sometimes appear in different notation. For instance, the covering number C(n, m, p) is defined as the minimum number of m-element subsets of X, needed to cover all p-element subsets (n > m > p). Obviously, C(n,m,p) = T(n,n -p,n -m).Let U(n, q, r) be the minimum number of subsets of Xn whose sizes are at least r and for which the size of any transversal (i.e. a subset intersecting each of them) is at least q. It is easy to see that U(n,q,r) = T(n,n -q + 1,r).(1) It was shown in I45] that T(n, k, r) is equal to the minimal length of a disjunctive normal form with n Boolean variables, which takes value 1 if at least k variables are equal to 1, and takes value 0 if less than r variables are equal to 1. Schrnheim [39], and independently, Katona, Nemetz and Simonovits [19] showed that I 1 r (n, k, r) > r(n -1, k, r) . Recursive Inequalitiesn~r

New constructions for covering designs

“…He also gave upper bounds and conjectures for T (n, 4, 3) and T (n, 5, 3), which stimulated much of the research. The results labeled 'Turán theory' in our tables either are described in recent survey papers by de Caen [8] and Sidorenko [29], or follow from constructions due to de Caen, Kreher, and Wiseman [10] or to Sidorenko [28].…”

“…For the rest: If t = 2, the lower bound is explained by Mills and Mullin [19] when it is less than 14 or has v ≤ 5, or explained by Todorov [34] [29].…”

On An Extremal Hypergraph Problem Of Brown, Erdős And Sós

“…In particular, when e = v r we get the well known Turán problem of determining the maximum possible number of edges in an r-graph that contains no complete r-graph on v vertices. See the surveys [8], [11], [14], and [21] for results and references on this and other graph and hypergraph Turán problems. In 1973, Brown, Erdős and Sós [5], [6] initiated the study of the function f for r-graphs (r ≥ 3).…”

On A Hypergraph Turán Problem Of Frankl

“…By choice of partition there are at least as many good edges containing a as bad. From (9) we see that a is in at least ways. Once X and Y have been chosen, to make E good b is constrained to lie in some particular class V i of the partition, so can be chosen in at most 1 2 + 1 10 η n ways.…”

“…numbers ex(n, F) when r > 2 is notoriously difficult, and exact results on hypergraph Turán numbers are very rare (see [3,9] for surveys). In this paper we obtain such a result for a sequence of hypergraphs introduced by Frankl. Let C (2k) r be the 2k-uniform hypergraph obtained by letting P 1 , · · · , P r be pairwise disjoint sets of size k and taking as edges all sets P i ∪ P j with i = j.…”