Abstract
Given a polynomial ring
S
=
k
[
x
1
,
⋯
,
x
n
]
over a field
k
, 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–Lindström 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–Lindström theorem and the Frankl–Füredi–Kalai theorem. We also show that the
f
-vectors of
(
a
1
,
⋯
,
a
n
)
-colored simplicial complexes or multicomplexes are never characterized by “reverse-lexicographic” complexes or multicomplexes when
n
>
1
and
(
a
1
,
…
,
a
n
)
≠
(
1
,
…
,
1
)
.