A Generalization of a Combinatorial Theorem of Macaulay

Clements, G. F.; Lindström, Bernt

doi:10.1007/978-0-8176-4842-8_30

Cited by 37 publications

(67 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Our proof is different from the original proof of the Frankl-Füredi-Kalai Theorem, but similar to another proof of the theorem given by London [13]. FranklFüredi-Kalai used a combinatorial technique called shifting, while our proof is based on compression, a technique which was introduced by Macaulay [14] and used efficiently by Clements-Lindström [4]. This paper is organized as follows: In Section 2, we show that colored squarefree rings are Macaulay-Lex.…”

Section: Conjecture 12 (Mermin-peeva) Suppose That R = A/m Is Macaulmentioning

confidence: 90%

Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin

Murai

2009

J Algebr Comb

View full text Add to dashboard Cite

. . . , x n ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R = A/M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Füredi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus-M ideals have maximal Betti numbers when A/M is Macaulay-Lex.

show abstract

Section: Conjecture 12 (Mermin-peeva) Suppose That R = A/m Is Macaulmentioning

confidence: 90%

Betti numbers of lex ideals over some Macaulay-Lex rings

Mermin

Murai

2009

J Algebr Comb

View full text Add to dashboard Cite

show abstract

“…This result was further generalized by Clements and Lindström in [3]. Daykin [4,5] gave two simple proofs, and later Hilton [9] gave another one.…”

Section: Theorem 4 For a Positive Integers N K And A Set U Of N Setsmentioning

confidence: 91%

$$f$$ -Vectors Implying Vertex Decomposability

Lasoń

2012

Discrete Comput Geom

View full text Add to dashboard Cite

show abstract

“…Compressed ideals were used heavily in [CL,MP1,MP2]. We introduce Acompressed ideals, which are more general.…”

Section: A-compressionmentioning

confidence: 99%

“…Clements and Lindström [CL] proved that (x The main tool in our proofs is A-compression, which we introduce in Section 3. It generalizes the notion of x i -compression used in [CL,MP1,MP2].…”

Section: Introductionmentioning

confidence: 99%