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Ordered Ramsey numbers of powers of paths

António Girão, Barnabás Janzer, Oliver Janzer

Abstract

Given two vertex-ordered graphs $G$ and $H$, the ordered Ramsey number $R_<(G,H)$ is the smallest $N$ such that whenever the edges of a vertex-ordered complete graph $K_N$ are red/blue-coloured, then there is a red (ordered) copy of $G$ or a blue (ordered) copy of $H$. Let $P_n^t$ denote the $t$-th power of a monotone path on $n$ vertices. The ordered Ramsey numbers of powers of paths have been extensively studied. We prove that there exists an absolute constant $C$ such that $R_<(K_s,P_n^t)\leq R(K_s,K_t)^{C} \cdot n$ holds for all $s,t,n$, which is tight up to the value of $C$. As a corollary, we obtain that there is an absolute constant $C$ such that $R_<(K_n,P_n^t)\leq n^{Ct}$. These results resolve a problem and a conjecture of Gishboliner, Jin and Sudakov. Furthermore, we show that $R_<(P_n^t,P_n^t)\leq n^{4+o(1)}$ for any fixed $t$. This answers questions of Balko, Cibulka, Král and Kynčl, and of Gishboliner, Jin and Sudakov.

Ordered Ramsey numbers of powers of paths

Abstract

Given two vertex-ordered graphs and , the ordered Ramsey number is the smallest such that whenever the edges of a vertex-ordered complete graph are red/blue-coloured, then there is a red (ordered) copy of or a blue (ordered) copy of . Let denote the -th power of a monotone path on vertices. The ordered Ramsey numbers of powers of paths have been extensively studied. We prove that there exists an absolute constant such that holds for all , which is tight up to the value of . As a corollary, we obtain that there is an absolute constant such that . These results resolve a problem and a conjecture of Gishboliner, Jin and Sudakov. Furthermore, we show that for any fixed . This answers questions of Balko, Cibulka, Král and Kynčl, and of Gishboliner, Jin and Sudakov.
Paper Structure (9 sections, 12 theorems, 4 equations, 1 figure)

This paper contains 9 sections, 12 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation.
  3. Path power versus path power
  4. Path power versus clique
  5. The proof of Lemma \ref{['lemma_findKt+tfroms']}
  6. When $s$ is large compared to $t$
  7. When $s$ is not too large compared to $t$
  8. Concluding remarks and open problems
  9. Acknowledgements.

Key Result

Theorem 1.1

Let $t$ be a positive integer and $\varepsilon>0$. Then there is some $C=C(\varepsilon,t)>0$ such that, for all positive integers $n$, we have

Figures (1)

  • Figure 1: An example of an $(s,t)$-chain. Here the chain has length $k=3$, the striped regions $B_{i,j}$ form cliques of size $t=3$, and the sets $A_i$ have size at least $\left(\binom{s+t}{s}\right)^{10}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 3.1
  • Definition 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Lemma 3.6
  • ...and 5 more