András Gyárfás: Two Ramsey problems on vertex-ordered complete graphs inspired by twisted drawings

Two first-week exercises introducing Ramsey-type problems are:
1. In every 2-coloring of the edges of $K_n$ there is a monochromatic spanning tree.
2. In every 2-coloring of the edges of $K_{3n-1}$ there is a monochromatic matching of size n.
We discuss how these statements change for vertex-ordered complete graphs where independent edges can be nested, crossing or separated. Results and problems are from a recent joint work with János Barát and Géza Tóth.