1950 the 18-year old Edward Nelson posed the problem of finding Ï‡ (see its history in [S]). A number of relevant results were obtained under additional restrictions on monochromatic sets. K.â€¦ (More)

In [Covering a triangle with triangles, Amer. Math. Monthly 112 (1) (2005) 78; Cover-up, Geombinatorics XIV (1) (2004) 8â€“9], Conway and I showed that in order to cover an equilateral triangle of sideâ€¦ (More)

In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes onâ€¦ (More)

In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I formulated a conditional chromatic number theorem, which described aâ€¦ (More)

A six-coloring of the Euclidean plane is constructed such that the distance 1 is not realized by any color except one, which does not realize the distance x/2 1. A 44-year-old problem due to Edwardâ€¦ (More)