On sets without k-term arithmetic progression
نویسندگان
چکیده
منابع مشابه
On Sets of Integers Containing No k Elements in Arithmetic Progression
In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with b...
متن کاملOn the Density of Sets Containing No k-Element Arithmetic Progression of a Certain Kind
A theorem now known as Sperner’s Lemma [5] states that a largest collection of subsets of an n-element set such that no subset contains another is obtained by taking the collection of all the subsets with cardinal bn=2c. (We denote by bxc, resp. dxe, the largest integer less than or equal to x, resp. the smallest integer greater than or equal to x.) In other words, the density of a largest anti...
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 کاملThe number of subsets of integers with no k-term arithmetic progression
Addressing a question of Cameron and Erdős, we show that, for infinitely many values of n, the number of subsets of {1, 2, . . . , n} that do not contain a k-term arithmetic progression is at most 2O(rk(n)), where rk(n) is the maximum cardinality of a subset of {1, 2, . . . , n} without a k-term arithmetic progression. This bound is optimal up to a constant factor in the exponent. For all value...
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 ...
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ژورنال
عنوان ژورنال: Journal of Computer and System Sciences
سال: 2012
ISSN: 0022-0000
DOI: 10.1016/j.jcss.2011.09.003