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1.
For an l-graph
, the Turán number
is the maximum number of edges in an n-vertex l-graph
containing no copy of
. The limit
is known to exist [8]. The Ramsey–Turán density
is defined similarly to
except that we restrict to only those
with independence number o(n). A result of Erdős and Sós [3] states that
as long as for every edge E of
there is another edge E′of
for which |E∩E′|≥2. Therefore a natural question is whether there exists
for which
.
Another variant
proposed in [3] requires the stronger condition that every set of vertices of
of size at least εn (0<ε<1) has density bounded below by some threshold. By definition,
for every
. However, even
is not known for very many l-graphs
when l>2.
We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we
construct, for each l≥3, infinitely many l-graphs
for which
.
We also prove that the 3-graph
with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies
. The existence of a hypergraph
satisfying
was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based
on different ideas and is simpler than these earlier proofs.
* Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P.
Sloan Research Fellowship.
† Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529. 相似文献
2.
Abstract
By
we denote the set of all propositional formulas. Let
be the set of all clauses. Define
. In Sec. 2 of this paper we prove that for normal modal logics
, the notions of
-expansions and
-expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of
-expansions for I and
-expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of
-expansions for I is the same as that of
-expansions for I.
The project is supported by NSFC 相似文献
3.
4.
Let n and r be positive integers. Suppose that a family
satisfies F1∩···∩Fr ≠∅ for all F1, . . .,Fr ∈
and
. We prove that there exists ε=ε(r) >0 such that
holds for 1/2≤w≤1/2+ε if r≥13. 相似文献
5.
We define the reduced minimum modulus
of a nonzero element a in a unital C
*-algebra
by
. We prove that
. Applying this result to
and its closed two side ideal
, we get that dist
,
and
for any
if RR
= 0, where
and
is the quotient homomorphism and
. These results generalize corresponding results in Hilbert spaces. 相似文献
6.
Michael Capalbo 《Combinatorica》2005,25(4):379-391
Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite
family
of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈
satisfy the following properties. The output-set Y has cardinality
times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of
is simple to specify, and each
has fewer than
edges. We then modify
to obtain explicit unique-neighbor concentrators of maximum degree 3.
* Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013. 相似文献
7.
Let E be a non empty set, let P : = E × E,
:= {x × E|x ∈ E},
:= {E × x|x ∈ E}, and
:= {C ∈ 2
P
|∀X ∈
: |C ∩ X| = 1} and let
. Then the quadruple
resp.
is called chain structure resp. maximal chain structure. We consider the maximal chain structure
as an envelope of the chain structure
. Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes.
Here we study in detail the groups of automorphisms
,
,
,
related to a maximal chain structure
. The set
of all chains can be turned in a group
such that the subgroup
of
generated by
the left-, by
the right-translations and by ι the inverse map of
is isomorphic to
(cf. (2.14)). 相似文献
8.
A partial tube in PG(3, q) is a pair
, where
is a collection of mutually disjoint lines of PG(3, q) with the property that for each plane π of PG(3, q) through L, the intersection of π with the lines of
is an arc. Here, we generalize the notion of partial tube allowing the ground field to be any algebraically closed field.
To a generalized partial tube we will associate an irreducible surface of degree d in
providing upper bounds on d.
The authors were partially supported by MIUR and GNSAGA of INdAM (Italy). 相似文献
9.
Mark Pankov 《Journal of Geometry》2004,79(1-2):169-176
Let
be a finite-dimensional projective space
and
be the Grassmannian consisting of
all k-dimensional subspaces of
. In the paper we show that
transformations of
sending base subsets
to base subsets are induced by collineations of
to itself or to the dual projective space
.
This statement generalizes the main result of the authors paper [19]. 相似文献
10.
Let
be the 2k-uniform hypergraph obtained by letting P1, . . .,Pr be pairwise disjoint sets of size k and taking as edges all sets Pi∪Pj with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph Kr: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain
can be obtained by partitioning V = V1 ∪V2 and taking as edges all sets of size 2k that intersect each of V1 and V2 in an odd number of elements. Let
denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no
has at most as many edges as
.
Sidorenko has given an upper bound of
for the Tur′an density of
for any r, and a construction establishing a matching lower bound when r is of the form 2p+1. In this paper we also show that when r=2p+1, any
-free hypergraph of density
looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density
of
to
, where c(r) is a constant depending only on r.
The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections
in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear
algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials.
* Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship. 相似文献
11.
A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and
. Let
be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have
This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi.
We also show that equality holds if and only if
consists of all r-element subsets containing a fixed element.
Analogous results are obtained for nonuniform families. 相似文献
12.
In the late 1950s, B. Segre introduced the fundamental
notion of arcs and complete arcs [48,49]. An arc in a nite
projective plane is a set of points with no three on a line and
it is complete if cannot be extended without violating this
property. Given a projective plane
, determining
, the size of its
smallest complete arc, has been a major open question in finite
geometry for several decades. Assume that
has order
q, it was shown by Lunelli
and Sce [41], more than 40 years ago, that
. Apart from this bound,
practically nothing was known about
, except for the case
is the Galois plane. For
this case, the best upper bound, prior to this paper, was
O(q
3/4)
obtained by Sznyi using the properties of the Galois field
GF(q).In this paper, we prove that
for any projective plane
of order
q, where
c is a universal constant.
Together with Lunelli-Sces lower bound, our result determines
up to a polylogarithmic
factor. Our proof uses a probabilistic method known as the
dynamic random construction or Rödls nibble. The proof also
gives a quick randomized algorithm which produces a small
complete arc with high probability.The key ingredient of our proof is a new concentration
result, which applies for non-Lipschitz functions and is of
independent interest.* Research supported in part by grant RB091G-VU from
UCSD, by NSF grant DMS-0200357 and by an A. Sloan
fellowship.Part of this work was done at AT&T Bell Labs and
Microsoft Research 相似文献
13.
We consider Dirichlet spaces (
) in L
2 and more general energy forms
in L
p
,
. For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that
, resp.
, are compactly embedded in L
2, resp. L
p
, we prove a Poincaré inequality for transient (Dirichlet) forms. If both
and its adjoint
are sub-Markovian semigroups, we show that the transience of T
t
is independent of
) and that it is implied by the transience of the energy form
of
and the form
belonging to
. 相似文献
14.
Stefan Gille 《Archiv der Mathematik》2007,88(4):333-343
Let
be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex
then there exists a dualizing complex
of Z such that there is an isomorphism of coherent Witt groups
for all
.
Received: 3 March 2006 相似文献
15.
We show that the hereditary discrepancy of a hypergraph
on n points increases by a factor of at most O(log n) when one adds a new edge to
. 相似文献
16.
A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper,
we show, for a weakly closed linear subspace
of a CDCSL algebra
, that
is a Lie ideal if and only if
for all invertibles A in
, and that
is a Jordan ideal if and only if it is an associative ideal. 相似文献
17.
The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy
of evolution families on the real line. We consider two general classes denoted
and
and we prove that if V,W are Banach function spaces with
and
, then the admissibility of the pair
for an evolution family
implies the uniform dichotomy of
. In addition, we consider a subclass
and we prove that if
, then the admissibility of the pair
implies the uniform exponential dichotomy of the family
. This condition becomes necessary if
. Finally, we present some applications of the main results. 相似文献
18.
For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved
for even n and
, respectively. Recently, Johann confirmed the following conjectures of Hendry:
for
and kn even and
for n = 2kq, where q is a positive integer. In this paper we prove
for
and kn even, and we determine m(n, 3). 相似文献
19.
Alina Iacob 《Archiv der Mathematik》2005,85(4):335-344
We consider two pairs of complete hereditary cotorsion theories
on the category of left R-modules, such that
We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules
can be computed either using a left
of M and a left
of M or using a right
a right
of N.
Received: 17 December 2004 相似文献
20.
For an arbitrary set E and a given closure operator
, we want to construct a symmetric closure operator
via some – possibly infinite – iteration process. If E is finite, the corresponding symmetric closure operator .
defines a matroid. If
and
is the convex closure operator,
turns out to be the affine closure operator. Moreover, we apply the symmetrization process to closure operators induced by
visibility.
Received March 9, 2005 相似文献