| Full text | |
| Author(s): |
Total Authors: 4
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| 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
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| 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 Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 17/02263-0 - Property testing and estimation of graph parameters |
| Grantee: | Henrique Stagni |
| Support Opportunities: | 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 Opportunities: | Scholarships in Brazil - Doctorate |