527 research outputs found
A note on long rainbow arithmetic progressions
Jungi\'{c} et al (2003) defined as the minimal number such that there is a rainbow arithmetic progression of length
in every equinumerous -coloring of for every .
They proved that for every , and conjectured that . We prove
for all that using the
K\H{o}v\'{a}ri-S\'{o}s-Tur\'{a}n theorem and Wigert's bound on the divisor
function.Comment: 3 page
The structure of rainbow-free colorings for linear equations on three variables in Zp
Let p be a prime number and Zp be the cyclic group of order p. A coloring of
Zp is called rainbow-free with respect to a certain equation, if it contains no
rainbow solution of the same, that is, a solution whose elements have pairwise
distinct colors. In this paper we describe the structure of rainbow-free
3-colorings of Zp with respect to all linear equations on three variables.
Consequently, we determine those linear equations on three variables for which
every 3-coloring (with nonempty color classes) of Zp contains a rainbow
solution of it
Rainbow-free 3-colorings of Abelian Groups
A 3-coloring of the elements of an abelian group is said to be rainbow--free
if there is no 3-term arithmetic progression with its members having pairwise
distinct colors. We give a structural characterization of rainbow--free
colorings of abelian groups. This characterization proves a conjecture of
Jungi\'c et al. on the size of the smallest chromatic class of a rainbow-free
3-coloring of cyclic groups.Comment: 19 page
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
Counting patterns in colored orthogonal arrays
Let be an orthogonal array and let be an --coloring of
its ground set . We give a combinatorial identity which relates the number
of vectors in with given color patterns under with the cardinalities of
the color classes. Several applications of the identity are considered. Among
them, we show that every equitable --coloring of the integer interval
has at least monochromatic Schur triples. We also
show that in an orthogonal array , the number of monochromatic
vectors of each color depends only on the number of vectors which miss that
color and the cardinality of the color class.Comment: 12 page
Long rainbow arithmetic progressions
Define as the minimal for which there is a rainbow
arithmetic progression of length in every equinumerous -coloring of
for all . Jungi\'{c}, Licht (Fox), Mahdian, Nesetril
and Radoici\'{c} proved that . We almost
close the gap between the upper and lower bounds by proving that . Conlon, Fox and Sudakov have independently shown
a stronger statement that .Comment: Minor revisions, to appear in Journal of Combinatoric
Anti-van der Waerden Numbers of Graph Products
In this paper, anti-van der Waerden numbers on Cartesian products of graphs
are investigated and a conjecture made by Schulte, et al (see arXiv:1802.01509)
is answered. In particular, the anti-van der Waerden number of the Cartesian
product of two graphs has an upper bound of four. This result is then used to
determine the anti-van der Waerden number for any Cartesian product of two
paths.Comment: 15 pages, 3 figure
Rainbow Arithmetic Progressions in Finite Abelian Groups
For positive integers and , the \emph{anti-van der Waerden number} of
, denoted by , is the minimum number of
colors needed to color the elements of the cyclic group of order and
guarantee there is a rainbow arithmetic progression of length . Butler et
al. showed a reduction formula for in terms of the
prime divisors of . In this paper, we analagously define the anti-van der
Waerden number of a finite abelian group and show is determined
by the order of and the number of groups with even order in a direct sum
isomorphic to . The \emph{unitary anti-van der Waerden number} of a group is
also defined and determined
Rainbow numbers of for
Consider the set and an equation . The rainbow
number of for , denoted , is the smallest
number of colors such that for every exact -coloring of , there exists a solution to with every member of
the solution set assigned a distinct color. This paper focuses on linear
equations and, in particular, establishes the rainbow number for the equations
for and . The paper also establishes a
general lower bound for .Comment: 8 pages; added references, renamed the parameter 'rainbow number',
notation has been updated to reflect convention, revised the statement and
proof of Lemma 3.1, revised the proof of Proposition 3.4, results are
unchanged; updated language, updated formatting, results are unchange
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.\
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