On sets of integers containing k elements in 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 Sets of Integers Not Containing Long Arithmetic Progressions
After this paper was completed, we learned that the main result had in fact been proved much earlier by R.A. Rankin (”Sets of integers containing not more than a given number of terms in arithmetical progression”, Proc. Roy. Soc. Edinburgh Sect. A 65 (1960/1961), 332–344). Since very few people appear to have been aware of that result, I have decided to leave the present paper on my web page as...
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 sequences of positive integers containing no p terms in arithmetic progression
We use topological ideas to show that, assuming the conjecture of Erdös [4] on subsets of positive integers having no p terms in arithmetic progression (A. P.), there must exist a subset Mp of positive integers with no p terms in A. P. with the property that among all such subsets, Mp maximizes the sum of the reciprocals of its elements.
متن کاملOn Arithmetic Progressions in Sums of Sets of Integers
3 Proof of Theorem 1 9 3.1 Estimation of the g1 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Estimation of the g3 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.3 Estimation of the g2 term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Putting everything together. . . . . . . . . . . . . . . . . . . . . ....
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1975
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-27-1-199-245