This is the first in a series of three papers by these authors. They all are concerned with two open problems by Erdös. The first is to prove that for every non-equilateral triangle \(T\) and for any 2-coloring of a plane, there is a monochromatic copy of \(T\). The second problem asks for the smallest \(r\) such that a covering of the plane by \(r\) sets, with some specified properties, contains no two points a unit distance apart and both in the same set. The following are the two main results proved in the paper: (1) For every two-coloring of the plane the set of sizes of monochromatic equilateral triangles is uncountable. (2) For every four-coloring of the plane the set of reals \(d\) such that there exist two points at distance \(d\) and of the same color is uncountable.