The linear chromatic number of a Sperner family

Let SS be a finite set and SS a complete Sperner family on SS, i.e. a Sperner family such that every x∈Sx∈S is contained in some member of SS. The linear chromatic number of SS, defined by Cıvan, is the smallest integer nn with the property that there exists a function f:S→{1,…,n}f:S→{1,…,n} such that if f(x)=f(y)f(x)=f(y), then every set in SS which contains xx also contains yy or every set in SS which contains yy also contains xx. We give an explicit formula for the number of complete Sperner families on SS of linear chromatic number 22. We also prove tight bounds on the number of elements in a Sperner family of given chromatic number, and prove that complete Sperner families of maximum linear chromatic number are far more numerous those of lesser linear chromatic number.