HadwigerNelson Problem

The Chromatic Number of the Plane (CNP) is the chromatic number of the graph whose vertices are elements of the plane, and two points are connected by an edge if they are a unit distance apart. The HadwigerNelson problem asks to compute CNP. 
Unsolved
Polymath Project

4 
IMO 2009 Q6

Let $a_1, a_2, \ldots, a_n$ be distinct positive integers and let M be a set of $n1$ positive integers not containing $s = a_1 +a_2 +\ldots+a_n$. A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths $a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M. 
Polymath Project
Solved

0 
Twin Prime Conjecture

A twin prime is a prime number that is either 2 less or 2 more than another prime number. The Twin Prime Conjecture asks if there are infinitely many twin primes. 
Unsolved
Polymath Project

0 