I would like to write an article about powerful existence theorems that assert, under mild and simple conditions, that some basic pattern or regularity exist. See some examples below. By mild conditions I mean short, easy, general. By simple conditions I mean that they 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 -- thanks to the users that added more recent examples in the comments.

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

EDIT: I am adding great examples that were provided in the comments

(5) Van der Waerden's Theorem about arithmetic progression

(6) Max-Flow Min-Cut Theorem from graph theory

(7) Tverberg's Theorem about convex hulls

(8) A very recent one, Existence of Designs (Keevash, Glock et al.)