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(Reference retrieved automatically from Web of Science through information on FAPESP grant and its corresponding number as mentioned in the publication by the authors.)

ON SOME EXTREMAL RESULTS FOR ORDER TYPES

Author(s):
Han, J. [1] ; Kohayakawa, Y. [2] ; Sales, M. T. [3] ; Stagni, H. [2]
Total Authors: 4
Affiliation:
[1] Univ Rhode Isl, Dept Math, Kingston, RI 02881 - USA
[2] Univ Sao Paulo, Inst Matemat & Estat, Sao Paulo - Brazil
[3] Emory Univ, Dept Math, Atlanta, GA 30322 - USA
Total Affiliations: 3
Document type: Journal article
Source: ACTA MATHEMATICA UNIVERSITATIS COMENIANAE; v. 88, n. 3, p. 779-785, 2019.
Web of Science Citations: 0
Abstract

A configuration is a finite set of points in the plane. Two configurations A and B have the same order type if there exists a bijection between them preserving the orientation of every ordered triple. We investigate the following extremal problem on embedding configurations in general position in integer grid. Given an order type B, let ex(N, B) be the maximum integer m such that there exists a sub configuration of the integer grid {[}N](2) of size m without a copy of B. An application of the celebrated multidimensional Szemeredi's theorem gives ex(N, B) = o(N-2). We first prove a subquadratic upper bound for all large order types B and large N, namely, ex(N, B) <= N2-eta for some eta = eta(B) > 0. Then we give improved bounds for specific order types: we show that ex(N, B) = 0(N) for the convex order type B, and ex(N, B) = N-3/2 + (o(1)) for those B satisfying the so-called Erdos-Hajnal property. Our approach is to study the inverse problem, that is, the smallest No = No (a, B) such that every a proportion of {[}N-0](2) contains a copy of B. (AU)

FAPESP's process: 13/03447-6 - Combinatorial structures, optimization, and algorithms in theoretical Computer Science
Grantee:Carlos Eduardo Ferreira
Support type: Research Projects - Thematic Grants
FAPESP's process: 17/02263-0 - Property testing and estimation of graph parameters
Grantee:Henrique Stagni
Support type: Scholarships abroad - Research Internship - Doctorate
FAPESP's process: 15/15986-4 - Asymptotic Combinatorics with Applications in Property Testing and Parameters Estimation.
Grantee:Henrique Stagni
Support type: Scholarships in Brazil - Doctorate