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

Abstract

Given a polynomial ring $S = \Bbbk[x_1, \dots, x_n]$ over a field $\Bbbk$, and a monomial ideal $M$ of $S$, we say the quotient ring $R = S/M$ is Macaulay-Lex if for every graded ideal of $R$, there exists a lexicographic ideal of $R$ 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-Lindstr\"om 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-Lindstr\"om theorem and the Frankl-F\"uredi-Kalai theorem. We also show that the $f$-vectors of $(a_1, \dots, a_n)$-colored simplicial complexes or multicomplexes are never characterized by "reverse-lexicographic" complexes or multicomplexes when $n>1$ and $(a_1, \dots, a_n) \neq (1, \dots, 1)$.