TY - JOUR

T1 - Ramsey numbers and monotone colorings

AU - Balko, Martin

N1 - Funding Information:
The project leading to this application has received funding from European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme under grant agreement No. 678765. The author was also supported by the grant 1452/15 from Israel Science Foundation and by the grant no. 18-13685Y of the Czech Science Foundation (GAČR). The author also acknowledges the support of the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004).
Publisher Copyright:
© 2018 Elsevier Inc.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - For positive integers N and r≥2, an r-monotone coloring of ({1,…,N}r) is a 2-coloring by −1 and +1 that is monotone on the lexicographically ordered sequence of r-tuples of every (r+1)-tuple from ({1,…,N}r+1). Let R‾mon(n;r) be the minimum N such that every r-monotone coloring of ({1,…,N}r) contains a monochromatic copy of ({1,…,n}r). For every r≥3, it is known that R‾mon(n;r)≤towr−1(O(n)), where towh(x) is the tower function of height h−1 defined as tow1(x)=x and towh(x)=2towh−1(x) for h≥2. The Erdős–Szekeres Lemma and the Erdős–Szekeres Theorem imply R‾mon(n;2)=(n−1)2+1 and R‾mon(n;3)=(2n−4n−2)+1, respectively. It follows from a result of Eliáš and Matoušek that R‾mon(n;4)≥tow3(Ω(n)). We show that R‾mon(n;r)≥towr−1(Ω(n)) for every r≥3. This, in particular, solves an open problem posed by Eliáš and Matoušek and by Moshkovitz and Shapira. Using two geometric interpretations of monotone colorings, we show connections between estimating R‾mon(n;r) and two Ramsey-type problems that have been recently considered by several researchers. Namely, we show connections with higher-order Erdős–Szekeres theorems and with Ramsey-type problems for order-type homogeneous sequences of points. We also prove that the number of r-monotone colorings of ({1,…,N}r) is 2Nr−1/rΘ(r) for N≥r≥3, which generalizes the well-known fact that the number of simple arrangements of N pseudolines is 2Θ(N2).

AB - For positive integers N and r≥2, an r-monotone coloring of ({1,…,N}r) is a 2-coloring by −1 and +1 that is monotone on the lexicographically ordered sequence of r-tuples of every (r+1)-tuple from ({1,…,N}r+1). Let R‾mon(n;r) be the minimum N such that every r-monotone coloring of ({1,…,N}r) contains a monochromatic copy of ({1,…,n}r). For every r≥3, it is known that R‾mon(n;r)≤towr−1(O(n)), where towh(x) is the tower function of height h−1 defined as tow1(x)=x and towh(x)=2towh−1(x) for h≥2. The Erdős–Szekeres Lemma and the Erdős–Szekeres Theorem imply R‾mon(n;2)=(n−1)2+1 and R‾mon(n;3)=(2n−4n−2)+1, respectively. It follows from a result of Eliáš and Matoušek that R‾mon(n;4)≥tow3(Ω(n)). We show that R‾mon(n;r)≥towr−1(Ω(n)) for every r≥3. This, in particular, solves an open problem posed by Eliáš and Matoušek and by Moshkovitz and Shapira. Using two geometric interpretations of monotone colorings, we show connections between estimating R‾mon(n;r) and two Ramsey-type problems that have been recently considered by several researchers. Namely, we show connections with higher-order Erdős–Szekeres theorems and with Ramsey-type problems for order-type homogeneous sequences of points. We also prove that the number of r-monotone colorings of ({1,…,N}r) is 2Nr−1/rΘ(r) for N≥r≥3, which generalizes the well-known fact that the number of simple arrangements of N pseudolines is 2Θ(N2).

KW - Erdős–Szekeres Theorem

KW - Monotone coloring

KW - Monotone path

KW - Ordered Ramsey number

KW - Ramsey number

UR - http://www.scopus.com/inward/record.url?scp=85056829998&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2018.11.013

DO - 10.1016/j.jcta.2018.11.013

M3 - Article

AN - SCOPUS:85056829998

SN - 0097-3165

VL - 163

SP - 34

EP - 58

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

ER -