Hilbert functions of colored quotient rings and a generalization of the Clements---Lindström theorem

Given a polynomial ring $$S = \Bbbk [x_1, \dots , x_n]$$S=k[x1,?,xn] over a field $$\Bbbk $$k , and a monomial ideal $$M$$M of $$S$$S, we say the quotient ring $$R = S/M$$R=S/M is Macaulay-Lex if for every graded ideal of $$R$$R, there exists a lexicographic ideal of $$R$$R with the same Hilbert function. In this paper, we introduce a class of quotient rings with combinatorial significance, which we call colored quotient rings. This class of rings include Clements---Lindstrom rings and colored squarefree rings as special cases that are known to be Macaulay-Lex. We construct two new classes of Macaulay-Lex rings, characterize all colored quotient rings that are Macaulay-Lex, and give a simultaneous generalization of both the Clements---Lindstrom theorem and the Frankl---Furedi---Kalai theorem. We also show that the $$f$$f-vectors of $$(a_1, \dots , a_n)$$(a1,?,an)-colored simplicial complexes or multicomplexes are never characterized by "reverse-lexicographic" complexes or multicomplexes when $$n>1$$n>1 and $$(a_1, \ldots , a_n) \ne (1, \ldots , 1)$$(a1,?,an)?(1,?,1).