On the Geometric Ramsey Number of Outerplanar Graphs

We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on $2n$ vertices are bounded by $O(n^{3})$ and $O(n^{10})$, in the convex and general case, respectively. We then apply similar methods to prove an $n^{O(\log(n))}$ upper bound on the Ramsey number of a path with $n$ ordered vertices.

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