## Chapter43The Hidden Order: A Painting

###### Project by: Eugene Kim and Arianna Jaffer.

$\textbf{Summary:}$ Our project is an acrylic painting on canvas based on the quote:

Complete chaos is impossible.– Theodore S. Motzkin, Israeli–American mathematician.

For the painting, we are focusing on the pioneers and trailblazers of Ramsey theory, including Paul Erdős, Esther Klein, George Szekeres, and Endre Makai.

The setting is the park Városliget in Budapest, Hungary, where the group spent time working on the happy ending problem, which led to a rediscovery of Ramsey's theorem.

As part of our research, we communicated with Endre Makai's son, Dr. Endre Makai Jr., a Hungarian mathematician. Here is an excerpt from our email exchange with Dr. Makai.

Thanks for both of you for your e–mail messages. Yes, Endre Makai (Sr.) was my father. That circle of mathematicians were just about his age.

By chance, I have found a photo of my father, with his three sons (I am the oldest one). My sister is not on the photo.

By the way, my father and Paul Turan together made probably the first proof of the problem of Erdős and Szekeres in the case of nine points.

Namely, any nine points in the plane, no three lying on a line, contain as a subset all five vertices of a convex pentagon. They did not publish this result, but later there appeared several different proofs for this special case.

The general conjecture was that $2^n + 1$ points in the plane, no three on a line, contain as a subset all $n$ vertices of a convex $n$–gon. For $2^n$ points this is still not true, so this conjecture, if true, would be sharp. Now the conjecture is known for constant times $2^{n(1 + \varepsilon )}$ points, where the constant factor depends on $\varepsilon \text{,}$ and where $\varepsilon \gt 0$ is an anyhow small positive number. This is already quite close to the original conjecture. Also the case of $17$ points is proved, by Szekeres, by a large computer help.

If you think, I can ask my brothers, if some of them has a photo of younger age from my father.

Best regards Endre Makai, Jr.