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 especially interested 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 in more recent results. Thanks to the users that added great examples in the comments!

(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) Szemerédi's Regularity Lemma from extremal graph theory

(9) Recent results about bounded gaps between primes (e.g. Zhang)

(10) Another recent one, Existence of Designs (Keevash, Glock et al.)