I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that a minimum regular pattern always exists. By mild conditions I mean short, easy, broad conditions. By simple conditions I mean not requiring advanced mathematical education. The conditions and the statement should be accessible to undergraduate mathematics/science students.

I am interested mostly in low-dimensional examples which allow an easy graphical representation.

I have some obvious examples in mind (given below), but they are rather classical results that were established between 1900 and 1950, roughly speaking.

I would be interested to see examples that are more recent.

Classical examples I have in mind

(1) Lemma of Sperner and Brouwer Fixed Point Theorem (for $n=2$)

(2) Lemma of Tucker and Borsuk-Ulam Theorem (for $n=2$)

(3) Ramsey’s Theorem (for the simplest case of 6 edges)

(4) Wagner’s Theorem about Planar Graphs

I would be grateful if you could point me to more recent examples.