Results

**1 - 3**of**3**### 4 1 5 9 6

"... Let a be the first prime in A. P. (not necessarily positive), d the common difference, s the last prime in A * P., n the number of primes in A. P., and let the residue r be the smallest positive integer such that a = r (mod d); if we keep a constant and increase d, we may speak of a search limit on ..."

Abstract
- Add to MetaCart

Let a be the first prime in A. P. (not necessarily positive), d the common difference, s the last prime in A * P., n the number of primes in A. P., and let the residue r be the smallest positive integer such that a = r (mod d); if we keep a constant and increase d, we may speak of a search limit on d, designated by a,; if we keep d constant and increase a and s, we may speak of a search limit on s, designated by s. The standard magic square of order 3 with elements 1, 3, • • • , 9 and center element c = 5, may be defined as 8 3

### Sub-Ramsey numbers for Arithmetic Progressions and Schur Triples

, 2005

"... For a given positive integer k, sr(m, k) denotes the minimal positive integer such that every coloring of [n], n ≥ sr(m, k), that uses each color at most k times, yields a rainbow AP(m); that is, an m-term arithmetic progression, all of whose terms receive different colors. We prove that 17 15 m2 8 ..."

Abstract
- Add to MetaCart

For a given positive integer k, sr(m, k) denotes the minimal positive integer such that every coloring of [n], n ≥ sr(m, k), that uses each color at most k times, yields a rainbow AP(m); that is, an m-term arithmetic progression, all of whose terms receive different colors. We prove that 17 15 m2 8 k + O(1) ≤ sr(3, k) ≤ 7 k + O(1) and sr(m, 2)> ⌊ 2 ⌋, improving the previous bounds of Alon, Caro, and Tuza from 1989. Our new lower bound on sr(m, 2) immediately implies that for n ≤ m2 2, there exists a mapping φ: [n] → [n] without a fixed point such that for every AP(m) A in [n], the set A ∩φ(A) is not empty. We also propose the study of sub-Ramsey–type problems for linear equations other than x + y = 2z. For a given positive integer k, we define ss(k) to be the minimal positive integer n such that every coloring of [n], n ≥ ss(k), that uses each color at most k times, yields a rainbow solution to the Schur equation x+y = z. We prove

### Rainbow arithmetic progressions

, 2014

"... 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 ..."

Abstract
- Add to MetaCart

(Show Context)
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], k) = n1−o(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. In fact, aw(Z2m, 3) = 3 for any positive integer m. However, for k ≥ 4, the behavior is similar to the previous case, that is, aw(Zn, k) = n1−o(1).