Ordered Ramsey and Tur\'an numbers of alternating paths and their variants

TL;DR

This paper investigates ordered Ramsey and Turán numbers for alternating paths, establishing bounds and exact values, and explores related ordered paths to understand their combinatorial properties.

Contribution

It provides new bounds for the ordered Ramsey number of alternating paths and determines their exact ordered Turán number, advancing understanding of ordered graph extremal problems.

Findings

Proved that R_<(AP_n) ≤ (2 + √2/2 + o(1))n.

Determined the exact ordered Turán number of AP_n.

Studied the ordered Ramsey and Turán numbers of related ordered paths.

Abstract

An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the ordered Ramsey number $R_<(H)$ is the smallest integer $N$ such that every red/blue edge-colouring of $K_N^<$ contains a monochromatic copy of $H$ with vertices appearing in the same relative order as in $H$. Balko, Cibulka, Kr\'al, and Kyn\v{c}l asked whether, among all ordered paths on $n$ vertices, the ordered Ramsey number is minimised by the alternating path $\mathrm{AP}_n$ -- the ordered path with vertex set $[n]$ such that the vertices encountered along the path are $1, n, 2, n - 1,3, n-2,\dots$. Motivated by this problem, we make progress on establishing the value of $R_<(\mathrm{AP}_n)$ by proving that \[ R_{<}(\mathrm{AP}_n)\leq \left(2+\frac{\sqrt{2}}{2}+o(1)\right)n. \] We then use similar methods to determine the exact ordered Tur\'an number of $\mathrm{AP}_n$, and study the ordered Ramsey and Tur\'an numbers of several related ordered paths.