Rainbow arithmetic progressions in finite abelian groups
نویسندگان
چکیده
منابع مشابه
Rainbow Arithmetic Progressions in Finite Abelian Groups
For positive integers n and k, the anti-van der Waerden number of Zn, denoted by aw(Zn, k), is the minimum number of colors needed to color the elements of the cyclic group of order n and guarantee there is a rainbow arithmetic progression of length k. Butler et al. showed a reduction formula for aw(Zn, 3) = 3 in terms of the prime divisors of n. In this paper, we analagously define the anti-va...
متن کاملOn Subsets of Finite Abelian Groups with Arithmetic Progressions
Brown and Buhler [ 3 ] and F rank l , G r a h a m and R6dl [ 4 ] p roved tha t D(G) = o(IG I) for all G (here, and t h r o u g h o u t the pape r G denotes a finite abe l ian g roup of odd order) . In this no te we are in teres ted in D(G) for g roups with m a n y const i tuents . A lon and D u b i n e r [ 1, 2] asked whether there exists a cons tan t c < 3 such tha t D(Z~)~<c". I. Ruzsa has re...
متن کاملRainbow 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 {1, . . . , n} can be colored and still guarantee there is a rainbow arithmetic progression of length k. We establish that aw([n], 3) = Θ(log n) and aw([n...
متن کاملOn Rainbow Arithmetic Progressions
Consider natural numbers {1, · · · , n} colored in three colors. We prove that if each color appears on at least (n + 4)/6 numbers then there is a three-term arithmetic progression whose elements are colored in distinct colors. This variation on the theme of Van der Waerden’s theorem proves the conjecture of Jungić et al.
متن کاملOn rainbow 4-term arithmetic progressions
{sl Let $[n]={1,dots, n}$ be colored in $k$ colors. A rainbow AP$(k)$ in $[n]$ is a $k$ term arithmetic progression whose elements have different colors. Conlon, Jungi'{c} and Radoiv{c}i'{c} cite{conlon} prove that there exists an equinumerous 4-coloring of $[4n]$ which is rainbow AP(4) free, when $n$ is even. Based on their construction, we show that such a coloring of $[4n]$...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Combinatorics
سال: 2018
ISSN: 2156-3527,2150-959X
DOI: 10.4310/joc.2018.v9.n4.a3