On the Turán number of ordered forests

Abstract

An ordered graph H is a simple graph with a linear order on its vertex set. The corresponding Turán problem, first studied by Pach and Tardos, asks for the maximum number ex < (n, H) of edges in an ordered graph on n vertices that does not contain H as an ordered subgraph. It is known that ex < (n, H) > n 1+ε for some positive ε = ε(H) unless H is a forest that has a proper 2-coloring with one color class totally preceding the other one. Making progress towards a conjecture of Pach and Tardos, we prove that ex < (n, H) = n 1+o (1) holds for all such forests that are "degenerate" in a certain sense. This class includes every forest for which an n 1+o(1) upper bound was previously known, as well as new examples. Our proof is based on a density-increment argument.

Extremal Theory of Ordered Graphs

“…On the positive side, ex < (P; n) = O(n log c n) was established in Pach and Tardos [2006] for all ordered bipartite forests with at most 6 vertices. The most general result in this direction is due to Korándi, Tardos, Tomon, and Weidert [2017]. They call a 0-1 matrix M vertically degenerate if for any submatrix M 0 = (a ij ) of M consisting of l > 1 rows one can find 1 Ä k < l such that M 0 has at most one column j with two 1-entries a ij = a i 0 j = 1 satisfying 1 Ä i Ä k < i 0 Ä l. Note that all vertically degenerate 0-1 matrices are cycle-free.…”

Ordered and Convex Geometric Trees with Linear Extremal Function

“…Thus, i a < i f < i c . This means we need to consider only cyclical structure (1,5,3,4,2) and (when we switch from the case of odd i c to the even) (5,1,3,2,4).…”

“…A particularly interesting phenomenon, discovered by Füredi and Hajnal [3], is that the order of magnitude of the extremal function for the ordered forest {13, 35, 24, 46} consisting of two interlacing paths of length two is determined by the extremal theory for Davenport-Schinzel sequences, and in particular the extremal function has order of magnitude Θ(nα(n)), where α(n) is the inverse Ackermann function. Further progress towards the conjecture was made by Korándi, Tardos, Tomon and Weidert [5], in the equivalent reformulation of the problem in terms of forbidden 0-1 submatrices of 0-1 matrices, giving a wide class of graphs F for which ex → (n, F ) = n 1+o (1) as n → ∞. The following basic question closely related to Conjecture 1.1 also has information theoretic applications (see for instance [1]).…”

On the Turán number of some ordered even cycles

“…The problem of finding the extremal number of matrix patterns was introduced by Füredi and Hajnal [7] about 25 years ago, and several results have been obtained since then (see e.g. [12,15,16,18] and the references therein), although most of them concern matrices of acyclic graphs. One notable exception is a result of Pach and Tardos [16] that establishes ex < (n, H) = Θ(n 4/3 ) for an infinite set of ordered cycles H that they call "positive" cycles.…”