Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMultidimensional extension of the Morse--Hedlund theorem
226
0
0.0
(
0
)
Added by
Fabien Durand
Publication date
2011
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
A celebrated result of Morse and Hedlund, stated in 1938, asserts that a sequence $x$ over a finite alphabet is ultimately periodic if and only if, for some $n$, the number of different factors of length $n$ appearing in $x$ is less than $n+1$. Attempts to extend this fundamental result, for example, to higher dimensions, have been considered during the last fifteen years. Let $dge 2$. A legitimate extension to a multidimensional setting of the notion of periodicity is to consider sets of $ZZ^d$ definable by a first order formula in the Presburger arithmetic $<ZZ;<,+>$. With this latter notion and using a powerful criterion due to Muchnik, we exhibit a complete extension of the Morse--Hedlund theorem to an arbitrary dimension $d$ and characterize sets of $ZZ^d$ definable in $<ZZ;<,+>$ in terms of some functions counting recurrent blocks, that is, blocks occurring infinitely often.
rate research
Read More
A propositional logic sentence in conjunctive normal form that has clauses of length two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that has been reduced, that is, which is unchanged under application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of of the original graph. We conclude with a discussion of why the Robertson-Seymour graph minor theorem does not apply in our approach.
We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerdens proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.
The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyarfas-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets $mathcal F$ of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of $mathcal F$ as an induce subdigraph. Among notable results, we prove that oriented triangle-free graphs without a directed path of length $3$ are $2$-colorable. If condition of triangle-free is replaced with $K_4$-free, then we have an upper bound of $414$. We also show that an orientation of complete multipartite graph with no directed triangle is 2-colorable. To prove these results we introduce the notion of emph{nice sets} that might be of independent interest.
In [J. Combin. Theory Ser. B 70 (1997), 2-44] we gave a simplified proof of the Four-Color Theorem. The proof is computer-assisted in the sense that for two lemmas in the article we did not give proofs, and instead asserted that we have verified those statements using a computer. Here we give additional details for one of those lemmas, and we include the original computer programs and data as ancillary files accompanying this submission.
The Friedgut-Kalai-Naor (FKN) theorem states that if $f$ is a Boolean function on the Boolean cube which is close to degree 1, then $f$ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice. As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
