A distance graph is an undirected graph on the integers where two integers
are adjacent if their difference is in a prescribed distance set. The
independence ratio of a distance graph G is the maximum density of an
independent set in G. Lih, Liu, and Zhu [Star extremal circulant graphs, SIAM
J. Discrete Math. 12 (1999) 491--499] showed that the independence ratio is
equal to the inverse of the fractional chromatic number, thus relating the
concept to the well studied question of finding the chromatic number of
distance graphs.
We prove that the independence ratio of a distance graph is achieved by a
periodic set, and we present a framework for discharging arguments to
demonstrate upper bounds on the independence ratio. With these tools, we
determine the exact independence ratio for several infinite families of
distance sets of size three, determine asymptotic values for others, and
present several conjectures.Comment: 39 pages, 12 figures, 6 table
