Proof of the Finiteness of the Chromatic Number of Two-Dimensional Lacunary Distance Graphs
For mathematicians studying graph coloring and distance graphs, this provides a proof of finiteness for a class of two-dimensional graphs, extending known one-dimensional results.
The paper proves that any integer distance graph in two dimensions generated by a lacunary sequence of vectors has finite chromatic number, extending the one-dimensional lonely set method to two dimensions.
We extend the one-dimensional lonely set method to two dimensions for the purpose of studying the chromatic number of integer distance graphs in two dimensions. Given a lacunary sequence of displacement vectors in $Z^{2}$, we use a lacunary matrix theorem given by Broderick, Fishman and Kleinbock, to prove the existence of a satisfactory multiplier vector. We then give an explicit geometric colouring argument. This proves that any integer distance graph generated by a lacunary sequence of vectors in two dimensions has finite chromatic number.