Striking existence theorems with mild conditions, and simple to state: more recent examples? 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 had some obvious examples in mind (given below), but many of them are rather classical results established until around 1970, roughly speaking.
I would be interested in more recent results. Thanks to the users that added great examples in the comments!
(1) Cantor Set, and existence of cardinalities $>|\mathbb N |$
(2) Lemma of Sperner, and Brouwer Fixed Point Theorem
(3) Lemma of Tucker, and Borsuk-Ulam Theorem
(4) Ramsey's Theorem
(5) Wallpaper Groups: There exist exactly 17 plane symmetry groups
(6) Banach-Tarski Paradox
(7) Wagner's Theorem about Planar Graphs
(8) Monsky's Theorem
(9) Four Color Theorem
(10) Penrose Tiling
EDIT: adding great examples from the comments
(11) Max-Flow Min-Cut Theorem from graph theory
(12) Tverberg's Theorem about intersecting convex hulls
(13) Van der Waerden's Theorem
(14) Szemerédi's Regularity Lemma from extremal graph theory
(15) Recent results about Existence of Designs (Keevash 2014, Glock et al. 2016)
 A: There are a number of easily stated problems in elementary computational geometry that have been solved only relatively recently, e.g.,


*

*the origami existence theorem that a single rectangular sheet of paper can be folded into the shape of any connected polygonal region, even if it has holes;

*the fold-and-cut theorem that any shape with straight sides can be cut from a single sheet of paper by folding it flat and making a single straight complete cut;

*the carpenter's rule problem of moving a simple planar polygon continuously to a position where all its vertices are in convex position, without ever crossing itself (below is an example from Erik Demaine's website);


*the existence of hinged dissections; i.e., the existence of a common hinged dissection of any finite collection of polygons of equal area (below is an example due to Greg Frederickson).

A: adding other great examples, many of them provided in the comments section
(16) Kakeya needle problem and Besicovitch sets: You want to rotate a needle of unit length by $360°$. What is the region with smallest area to do that? It turns out there is no lower bound > 0 for the area of such a region, i.e. you can find arbitrarily small such regions. (https://en.wikipedia.org/wiki/Kakeya_set)
(17) A more recent one, Brenier’s Theorem on the existence of optimal transport maps between probability measures. (https://en.wikipedia.org/wiki/Transportation_theory_(mathematics))
(18) Recent results about bounded gaps between primes (e.g. Zhang)
(Adding these examples as an answer because the list of examples in my original question is getting too long)
A: 1) The set of continuous everywhere but differentiable nowhere functions on the unit interval is a meagre set of measure 1.
2) The existence of a space filling curve, or more generally surjective continuous maps $S^m \to S^n$ for $n>m$ (and then the fact that however any such map is homotopic to a map that misses a point).
A: Alexandrov's gluing theorem: If one glues polygons together along their boundaries to form a closed surface homeomorphic to a sphere, such that no point has more than $2\pi$ incident surface angle, then the result is 
isometric to a convex polyhedron, uniquely determined up to rigid motions. 
There is as yet no effective procedure to actually construct the polyhedron guaranteed
to exist.

A.D. Alexandrov. Convex Polyhedra. Springer-Verlag, Berlin, 2005. Monographs in Mathematics. Translation of the 1950 Russian ed. by N. S. Dairbekov,
  S.S. Kutateladze, and A.B. Sossinsky. p.100.


The result also holds for a single polygon, whose perimeter is glued closed 
by identifications:

          


          

Snapshots from a video by Erik Demaine, Martin Demaine, Anna Lubiw, J.O'Rourke, Irena Pashchenko.


A: This example is not really recent, since it was discovered in 1849 by Cayley and Salmon, but I think it qualifies. On a smooth cubic surface in $\mathbb{CP}^3$ there are exactly 27 lines.
This is a prototypical result in enumerative geometry. There are plenty of results in the same spirit, such as the 3264 conics tangent to 5 general conics, or the (related) 28 bitangents to a general quartic curve, but I think that Cayley-Salmon is striking for the simplicity of its hypothesis.
