Ordered Ramsey and Turán numbers of alternating paths and their variants

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, Král, and Kyn\v cl 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 Turán number of $\mathrm{AP}_n$, and study the ordered Ramsey and Turán numbers of several related ordered paths.

Figures (6)

• Figure 1: The four ordered paths on $8$ vertices of the form $P_8^{r_1,r_2}$.
• Figure 2: Above: The ordered path $\mathop{\mathrm{AP}}\nolimits_7$ and its matrix representation. Below: A red/blue edge-colouring of $K_6^<$ and its matrix representation. A monochromatic subgraph isomorphic to $\mathop{\mathrm{AP}}\nolimits_4$ (in fact, the unique one) is highlighted in green.
• Figure 3: An illustration of the algorithm from the proof of Theorem \ref{['th:ramsey']} with $n=8$, hence $N=17$, $A=[1,10]$ and $B=[8,17]$. The dark squares correspond to edges that are not between $A$ and $B$, while the grey squares correspond to the grey edges in the proof. If a cell contains a number, then this indicates the step in which the corresponding edge was deleted. The intervals $I_i$ and $J_i$ are highlighted. To reconstruct a monochromatic $\mathop{\mathrm{AP}}\nolimits_8$, start from any cell corresponding to an unremoved edge. Find a same-coloured edge on the same row that was removed in the $6$-th step, from which one can find a same-coloured edge in the same column removed in the $5$-th step, etc. For example, starting from the unremoved edge $10-11$, one can reconstruct a blue $\mathop{\mathrm{AP}}\nolimits_8$: $3-17-4-16-6-12-10-11$ (highlighted in the bottom right picture).
• Figure 4: On the left, an illustration of the algorithm from the proof of Theorem \ref{['th:turan']} applied with $N=17$ and $n=7$ to a graph with $56$ non-grey and $15$ grey edges. If a cell contains a number, then it indicates the step in which the corresponding edge was deleted. On the right, an illustration of the extremal graph for $N=17$ and $n=7$.
• Figure 5: An illustration of the grey edges from the proof of Proposition \ref{['prop:ramsey']} for $N=18$ and $n=8$. On the left is the case of $P_n^{<,<}$, on the right the case of $P_n^{>,>}$. The dark squares correspond to edges that are not between $A$ and $B$, while the grey ones correspond to the grey edges in the proof.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Corollary 3
• Corollary 4
• Proposition 5
• Theorem 6
• proof : Proof of Theorem \ref{['th:ramsey']}
• Claim 7
• proof
• Claim 8