79 research outputs found
Anti-van der Waerden numbers of 3-term arithmetic progressions
The \emph{anti-van der Waerden number}, denoted by , is the
smallest such that every exact -coloring of contains a rainbow
-term arithmetic progression. Butler et. al. showed that , and conjectured that
there exists a constant such that . In this paper, we show this conjecture is true by determining
for all . We prove that for ,
\[ aw([n],3)=\left\{\begin{array}{ll} m+2, & \mbox{if }\\ m+3, &
\mbox{otherwise}.
\end{array}\right.\
Rainbow Numbers of for
An exact -coloring of a set is a surjective function . The
rainbow number of a set for equation is the smallest integer such
that every exact -coloring of contains a rainbow solution to . In
this paper, the rainbow number of , for prime and the equation
is determined. The rainbow number of ,
for a natural number , is determined under certain conditions
Arithmetic Progressions in the Primitive Length Spectrum
In this article, we prove that every arithmetic locally symmetric orbifold of
classical type without Euclidean or compact factors has arbitrarily long
arithmetic progressions in its primitive length spectrum. Moreover, we show the
stronger property that every primitive length occurs in arbitrarily long
arithmetic progressions in its primitive length spectrum. This confirms one
direction of a conjecture of Lafont--McReynolds, which states that the property
of having every primitive length occur in arbitrarily long arithmetic
progressions characterizes the arithmeticity of such spaces
The ergodic and combinatorial approaches to Szemer\'edi's theorem
A famous theorem of Szemer\'edi asserts that any set of integers of positive
upper density will contain arbitrarily long arithmetic progressions. In its
full generality, we know of four types of arguments that can prove this
theorem: the original combinatorial (and graph-theoretical) approach of
Szemer\'edi, the ergodic theory approach of Furstenberg, the Fourier-analytic
approach of Gowers, and the hypergraph approach of Nagle-R\"odl-Schacht-Skokan
and Gowers. In this lecture series we introduce the first, second and fourth
approaches, though we will not delve into the full details of any of them. One
of the themes of these lectures is the strong similarity of ideas between these
approaches, despite the fact that they initially seem rather different.Comment: 48 pages, no figures. Based on a lecture series given at the Montreal
Workshop on Additive Combinatorics, April 6-12 200
On sequences covering all rainbow -progressions
Let denote the smallest positive integer with the property
that there exists an -colouring of such
that for every -subset there exists an
(arithmetic) -progression in with . Determining the behaviour of the function is a
previously unstudied problem. We use the first moment method to give an
asymptotic upper bound for for the case
An exploration of anti-van der Waerden numbers
In this paper results of the anti-van der Waerden number of various mathematical objects are discussed. The anti-van der Waerden number of a mathematical object G, denoted by aw(G,k), is the smallest r such that every exact r-coloring of G contains a rainbow k-term arithmetic progression. In this paper, results on the anti-van der Waerden number of the integers, groups such as the integers modulo n, and graphs are given. A connection between the Ramsey number of paths and the anti-van der Waerden number of graphs is established. The anti-van der Waerden number of [m]X[n] is explored. Finally, connections between anti-van der Waerden numbers, rainbow numbers, and anti-Schur numbers are discussed
Rainbow Arithmetic Progressions
In this paper, we investigate the anti-Ramsey (more precisely, anti-van der Waerden) properties of arithmetic progressions. For positive integers n and k, the expression aw([n]; k) denotes the smallest number of colors with which the integers f1; : : : ; ng can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n]; 3) = (log n) and aw([n]; k) = n1o(1) for k 4. For positive integers n and k, the expression aw(Zn; k) denotes the smallest number of colors with which elements of the cyclic group of order n can be colored and still guarantee there is a rainbow arithmetic progression of length k. In this setting, arithmetic progressions can \wrap around, and aw(Zn; 3) behaves quite differently from aw([n]; 3), depending on the divisibility of n. As shown in [Jungic et al., Combin. Probab. Comput., 2003], aw(Z2m; 3) = 3 for any positive integer m. We establish that aw(Zn; 3) can be computed from knowledge of aw(Zp; 3) for all of the prime factors p of n. However, for k 4, the behavior is similar to the previous case, that is, aw(Zn; k) = n1o(1)
Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory
The aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory
Exploiting Isomorphic Subgraphs in SAT (Long version)
While static symmetry breaking has been explored in the SAT community for
decades, only as of 2010 research has focused on exploiting the same discovered
symmetry dynamically, during the run of the SAT solver, by learning extra
clauses. The two methods are distinct and not compatible. The former prunes
solutions, whereas the latter does not -- it only prunes areas of the search
that do not have solutions, like standard conflict clauses. Both approaches,
however, require what we call \emph{full symmetry}, namely a
propositionally-consistent mapping between the literals, such that
, where here means syntactic
equivalence modulo clause ordering and literal ordering within the clauses. In
this article we show that such full symmetry is not a necessary condition for
adding extra clauses: isomorphism between possibly-overlapping subgraphs of the
colored incidence graph is sufficient. While finding such subgraphs is a
computationally hard problem, there are many cases in which they can be
detected a priory by analyzing the high-level structure of the problem from
which the CNF was derived. We demonstrate this principle with several
well-known problems, including Van der Waerden numbers, bounded model checking
and Boolean Pythagorean triples.Comment: The short version was submitted to SAT'2
Hitting sets and colorings of hypergraphs
In this paper we study the minimal size of edges in hypergraph families which
guarantees the existence of a polychromatic coloring, that is, a -coloring
of a vertex set such that every hyperedge contains a vertex of all color
classes. We also investigate the connection of this problem with -shallow
hitting sets: sets of vertices that intersect each hyperedge in at least one
and at most vertices.
We determine in some hypergraph families the minimal for which a
-shallow hitting set exists.
We also study this problem for a special hypergraph family, which is induced
by arithmetic progressions with a difference from a given set. We show
connections between some geometric hypergraph families and the latter, and
prove relations between the set of differences and polychromatic colorability
- …