I want to write an article about highly effective existence theorems that assert, underneath delicate and easy circumstances, {that a} minimal common sample all the time exists. By delicate circumstances I imply brief, simple, broad circumstances. By easy circumstances I imply not requiring superior mathematical training. The circumstances and the assertion must be accessible to undergraduate arithmetic/science college students.

I’m largely in low-dimensional examples which permit a straightforward graphical illustration.

I’ve some apparent examples in thoughts (given under), however they’re fairly classical outcomes that have been established between 1900 and 1950, roughly talking.
I’d have an interest to see examples which might be more moderen.

Classical examples I keep in mind

(1) Lemma of Sperner and Brouwer Mounted Level Theorem (for $n=2$)

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

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

(4) Wagner’s Theorem about Planar Graphs

I’d be grateful should you might level me to more moderen examples.