 
Summary: The Linear Chromatic Number of a Sperner Family
Reza Akhtar
Dept. of Mathematics
Miami University, Oxford, OH 45056, USA
reza@calico.mth.muohio.edu
Maxwell Forlini
Dept. of Mathematics
Miami University, Oxford, OH 45056, USA
forlinms@muohio.edu
July 29, 2010
Mathematics Subject Classification: 05D05
Abstract
Let S be a finite set and S a complete Sperner family on S, i.e. a Sperner
family such that every x S is contained in some member of S. The linear
chromatic number of S, defined by Civan, is the smallest integer n with the
property that there exists a function f : S {1, . . . , n} such that if f(x) =
f(y), then every set in S which contains x also contains y or every set in S
which contains y also contains x. We give an explicit formula for the number
of complete Sperner families on S of linear chromatic number 2. We also prove
tight bounds on the number of elements in a Sperner family of given chromatic
