256,234 research outputs found
"Weak yet strong'' restrictions of Hindman's Finite Sums Theorem
We present a natural restriction of Hindman’s Finite Sums Theorem that admits a simple combinatorial proof (one that does not also prove the full Finite Sums Theorem) and low computability-theoretic and proof-theoretic upper bounds, yet implies the existence of the Turing Jump, thus realizing the only known lower bound for the full Finite Sums Theorem. This is the first example of this kind. In fact we isolate a rich family of similar restrictions of Hindman’s Theorem with analogous propertie
Simplifications to A New Approach to the Covering Radius...”
We simplify the proofs of four results in [3], restating two of them for greater clarity. The main purpose of this note is to give a brief transparent proof of Theorem 7 of [3], the main upper bound of that paper. The secondary purpose is to give a more direct statement and proof of the integer programming determination of covering radius of [3]. Theorem 7 of [3] follows from a simple result in [2], which we state with the notation (for the linear code A)
Dimension and cut vertices: an application of Ramsey theory
Motivated by quite recent research involving the relationship between the
dimension of a poset and graph-theoretic properties of its cover graph, we show
that for every , if is a poset and the dimension of a subposet
of is at most whenever the cover graph of is a block of the cover
graph of , then the dimension of is at most . We also construct
examples which show that this inequality is best possible. We consider the
proof of the upper bound to be fairly elegant and relatively compact. However,
we know of no simple proof for the lower bound, and our argument requires a
powerful tool known as the Product Ramsey Theorem. As a consequence, our
constructions involve posets of enormous size.Comment: Final published version with updated reference
Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II
Let be a symmetric diffusion operator
with an invariant measure on a complete Riemannian
manifold. In this paper we prove Li-Yau gradient estimates for weighted
elliptic equations on the complete manifold with
and -dimensional Bakry-\'{E}mery Ricci curvature bounded below by some
negative constant. Based on this, we give an upper bound on the first
eigenvalue of the diffusion operator on this kind manifold, and thereby
generalize a Cheng's result on the Laplacian case (Math. Z., 143 (1975)
289-297).Comment: Final version. The original proof of Theorem 2.1 using Li-Yau
gradient estimate method has been moved to the appendix. The new proof is
simple and direc
On the Monotone Upper Bound Problem
The Monotone Upper Bound Problem asks for the maximal number M(d,n) of
vertices on a strictly-increasing edge-path on a simple d-polytope with n
facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n)
provided by McMullen's (1970) Upper Bound Theorem is tight, where M_{ubt}(d,n)
is the number of vertices of a dual-to-cyclic d-polytope with n facets.
It was recently shown that the upper bound M(d,n)<=M_{ubt}(d,n) holds with
equality for small dimensions (d<=4: Pfeifle, 2003) and for small corank
(n<=d+2: G\"artner et al., 2001). Here we prove that it is not tight in
general: In dimension d=6 a polytope with n=9 facets can have M_{ubt}(6,9)=30
vertices, but not more than 26 <= M(6,9) <= 29 vertices can lie on a
strictly-increasing edge-path.
The proof involves classification results about neighborly polytopes, Kalai's
(1988) concept of abstract objective functions, the Holt-Klee conditions
(1998), explicit enumeration, Welzl's (2001) extended Gale diagrams, randomized
generation of instances, as well as non-realizability proofs via a version of
the Farkas lemma.Comment: 15 pages; 6 figure
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