2,872 research outputs found
Hamilton paths with lasting separation
We determine the asymptotics of the largest cardinality of a set of Hamilton
paths in the complete graph with vertex set [n] under the condition that for
any two of the paths in the family there is a subpath of length k entirely
contained in only one of them and edge{disjoint from the other one
Maximum size of reverse-free sets of permutations
Two words have a reverse if they have the same pair of distinct letters on
the same pair of positions, but in reversed order. A set of words no two of
which have a reverse is said to be reverse-free. Let F(n,k) be the maximum size
of a reverse-free set of words from [n]^k where no letter repeats within a
word. We show the following lower and upper bounds in the case n >= k: F(n,k)
\in n^k k^{-k/2 + O(k/log k)}. As a consequence of the lower bound, a set of
n-permutations each two having a reverse has size at most n^{n/2 + O(n/log n)}.Comment: 10 page
Interlocked permutations
The zero-error capacity of channels with a countably infinite input alphabet
formally generalises Shannon's classical problem about the capacity of discrete
memoryless channels. We solve the problem for three particular channels. Our
results are purely combinatorial and in line with previous work of the third
author about permutation capacity.Comment: 8 page
Improving bounds on packing densities of 4-point permutations
We consolidate what is currently known about packing densities of 4-point
permutations and in the process improve the lower bounds for the packing
densities of 1324 and 1342. We also provide rigorous upper bounds for the
packing densities of 1324, 1342, and 2413. All our bounds are within
of the true packing densities. Together with the known bounds, this gives us a
fairly complete picture of all 4-point packing densities. We also provide new
upper bounds for several small permutations of length at least five. Our main
tool for the upper bounds is the framework of flag algebras introduced by
Razborov in 2007.Comment: journal style, 18 page
Exact Bounds for Some Hypergraph Saturation Problems
Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G
on vertex sets X,Y that satisfies the following condition; one can add the
edges between X and Y that do not belong to G one after the other so that
whenever a new edge is added, a new copy of K_{p,q} is created. The problem of
bounding W_n(p,q), and its natural hypergraph generalization, was introduced by
Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to
graphs, used algebraic methods to determine W_n(1,q).
Our main results in this paper give exact bounds for W_n(p,q), its hypergraph
analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In
particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n
then
W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2.
Our proof applies a reduction to a multi-partite version of the Two Families
theorem obtained by Alon. While the reduction is combinatorial, the main idea
behind it is algebraic
De Finetti theorem on the CAR algebra
The symmetric states on a quasi local C*-algebra on the infinite set of
indices J are those invariant under the action of the group of the permutations
moving only a finite, but arbitrary, number of elements of J. The celebrated De
Finetti Theorem describes the structure of the symmetric states (i.e.
exchangeable probability measures) in classical probability. In the present
paper we extend De Finetti Theorem to the case of the CAR algebra, that is for
physical systems describing Fermions. Namely, after showing that a symmetric
state is automatically even under the natural action of the parity
automorphism, we prove that the compact convex set of such states is a Choquet
simplex, whose extremal (i.e. ergodic w.r.t. the action of the group of
permutations previously described) are precisely the product states in the
sense of Araki-Moriya. In order to do that, we also prove some ergodic
properties naturally enjoyed by the symmetric states which have a
self--containing interest.Comment: 23 pages, juornal reference: Communications in Mathematical Physics,
to appea
Families of locally separated Hamilton paths
We improve by an exponential factor the lower bound of K¨orner and Muzi for the cardinality of the largest family of Hamilton paths in a complete graph of n vertices in which the union of any two paths has maximum degree 4. The improvement is through an explicit construction while the previous bound was obtained by a greedy algorithm. We solve a similar problem for permutations up to an exponential factor
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