249 research outputs found
Shadows and intersections in vector spaces
AbstractWe prove a vector space analog of a version of the Kruskal–Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erdős–Ko–Rado theorem for vector spaces
Shadows and intersections: stability and new proofs
We give a short new proof of a version of the Kruskal-Katona theorem due to
Lov\'asz. Our method can be extended to a stability result, describing the
approximate structure of configurations that are close to being extremal, which
answers a question of Mubayi. This in turn leads to another combinatorial proof
of a stability theorem for intersecting families, which was originally obtained
by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi
by means of a purely combinatorial result of Frankl. We also give an algebraic
perspective on these problems, giving yet another proof of intersection
stability that relies on expansion of a certain Cayley graph of the symmetric
group, and an algebraic generalisation of Lov\'asz's theorem that answers a
question of Frankl and Tokushige.Comment: 18 page
On f- and h- vectors of relative simplicial complexes
A relative simplicial complex is a collection of sets of the form , where are simplicial complexes.
Relative complexes played key roles in recent advances in algebraic, geometric,
and topological combinatorics but, in contrast to simplicial complexes, little
is known about their general combinatorial structure. In this paper, we address
a basic question in this direction and give a characterization of -vectors
of relative (multi)complexes on a ground set of fixed size. On the algebraic
side, this yields a characterization of Hilbert functions of quotients of
homogeneous ideals over polynomial rings with a fixed number of indeterminates.
Moreover, we characterize -vectors of fully Cohen--Macaulay relative
complexes as well as -vectors of Cohen--Macaulay relative complexes with
minimal faces of given dimensions. The latter resolves a question of Bj\"orner.Comment: accepted for publication in Algebraic Combinatoric
Quasipolynomial size frege proofs of Frankl's Theorem on the trace of sets
We extend results of Bonet, Buss and Pitassi on Bondy's Theorem and of Nozaki, Arai and Arai on Bollobas' Theorem by proving that Frankl's Theorem on the trace of sets has quasipolynomial size Frege proofs. For constant values of the parameter t, we prove that Frankl's Theorem has polynomial size AC(0)-Frege proofs from instances of the pigeonhole principle.Peer ReviewedPostprint (author's final draft
A Generalized Macaulay Theorem and Generalized Face Rings
We prove that the -vector of members in a certain class of meet
semi-lattices satisfies Macaulay inequalities. We construct a large family of
meet semi-lattices belonging to this class, which includes all posets of
multicomplexes, as well as meet semi-lattices with the "diamond property",
discussed by Wegner, as spacial cases. Specializing the proof to that later
family, one obtains the Kruskal-Katona inequalities and their proof as in
Wegner's.
For geometric meet semi lattices we construct an analogue of the exterior
face ring, generalizing the classic construction for simplicial complexes. For
a more general class, which include also multicomplexes, we construct an
analogue of the Stanley-Reisner ring. These two constructions provide algebraic
counterparts (and thus also algebraic proofs) of Kruskal-Katona's and
Macaulay's inequalities for these classes, respectively.Comment: Final version: 13 pages, 2 figures. Improved presentation, more
detailed proofs, same results. To appear in JCT
- …