## Abstract

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)≤tow_{r−1}(O(n)), where tow_{h}(x) is the tower function of height h−1 defined as tow_{1}(x)=x and tow_{h}(x)=2^{towh−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)≥tow_{3}(Ω(n)). We show that R‾_{mon}(n;r)≥tow_{r−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 2^{Nr−1/rΘ(r) } for N≥r≥3, which generalizes the well-known fact that the number of simple arrangements of N pseudolines is 2^{Θ(N2)}.

Original language | English |
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Pages (from-to) | 34-58 |

Number of pages | 25 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 163 |

DOIs | |

State | Published - 1 Apr 2019 |

## Keywords

- Erdős–Szekeres Theorem
- Monotone coloring
- Monotone path
- Ordered Ramsey number
- Ramsey number

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics