The length of an s-increasing sequence of r-tuples

Abstract

We prove a number of results related to a problem of Po-Shen Loh [9], which is equivalent to a problem in Ramsey theory. Let a = (a1, a2, a3) and b = (b1, b2, b3) be two triples of integers. Define a to be 2-less than b if a i < b i for at least two values of i, and define a sequence a1, …, a m of triples to be 2-increasing if a r is 2-less than a s whenever r < s. Loh asks how long a 2-increasing sequence can be if all the triples take values in {1, 2, …, n}, and gives a log* improvement over the trivial upper bound of n2 by using the triangle removal lemma. In the other direction, a simple construction gives a lower bound of n3/2. We look at this problem and a collection of generalizations, improving some of the known bounds, pointing out connections to other well-known problems in extremal combinatorics, and asking a number of further questions.

More Turán-Type Theorems for Triangles in Convex Point Sets

“…However, this bound is not tight. A sequence of recursive constructions has steadily raised this lower bound [6,10,11,12,13]. The current record is held by Gowers and Long [6], who describe a construction of size Ω(n 1.546 ).…”

Extremal problems for pairs of triangles

“…The problem of determining ex * (n, S 2 ) appears to be very difficult, as it is connected to monotone matrices, tripod packing, and 2-comparable sets -see Aronov, Dujmović, Morin, Ooms and da Silveira [1] for details. The best bounds are ex * (n, S 2 ) = Ω(n 1.546 ) due to Gowers and Long [12] and ex * (n, S 2 ) = n 2 / exp(Ω(log * n)) due to the best bounds on the removal lemma by Fox [7].…”

“…In particular, the extremal problem for forbidding both S 2 and D 2 is tightly connected to the monotone matrix problem, to the tripod packing problem and 2-comparable triples problem. Lower bounds on the extremal function in these cases were given by Gowers and Long [12], and upper bounds come from the triangle removal lemma, however these bounds remain far apart. Extremal problems for matchings in ordered graphs connect to enumeration of permutations [19] and extensions to hypergraphs [15].…”

Tight paths in convex geometric hypergraphs

“…Given an r-uniform cgh F, let ex (n, F) denote the maximum number of edges in an n-vertex r-uniform cgh that does not contain F. Extremal problems for convex geometric graphs (or cggs for short) have been studied extensively, going back to theorems in the 1930's on disjoint line segments in the plane. We refer the reader to the papers of Braß, Károlyi and Valtr [3], Capoyleas and Pach [5] and the references therein for many related extremal problems on convex geometric graphs and to Aronov, Dujmovič, Morin, Ooms and da Silveira [1], Braß [2], Brass, Rote and Swanepoel [4], and Pach and Pinchasi [17] for problems in convex geometric hypergraphs, and their connections to important problems in discrete geometry, as well as the triangle-removal problem (see Aronov, Dujmovič, Morin, Ooms and da Silveira [1] and Gowers and Long [11]).…”