Four unsolved mathematical problems (Christmas Lecture 2015) [English subtitles] 

The Hadwiger-Nelson problem, Ramsey numbers, the Toeplitz conjecture and the Busy Beaver problem.
Some progress regarding the first problem: arxiv.org/abs/1804.02385v3
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00:00 Introduction
12:20 Hadwiger-Nelson Problem
21:26 Ramsey Numbers
42:02 Toeplitz Conjecture
59:14 Busy Beavers