Michael Young (Iowa State University) Coloring the Integers with Rainbow Arithmetic Progressions
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1 Coloring the Integers with Rainbow Arithmetic Progressions Michael Young Iowa State University
2 A k-term arithmetic progression is a finite sequence of k terms of the form {a, a + d, a + 2d,..., a + (k 1)d}, where k, a, and d are nonnegative integers. 3, 10, 17, 24, 31
3 anti-van der Waerden numbers An exact r-coloring of a set S is a function c : S C, such that C = r. 3, 10, 17, 24, 31
4 anti-van der Waerden numbers An exact r-coloring of a set S is a function c : S C, such that C = r. 3, 10, 17, 24, 31 A set S is rainbow under an r-coloring c, if c(s 1 ) c(s 2 ), for each distinct s 1, s 2 S.
5 anti-van der Waerden numbers Given positive integers n and k with k n, the anti-van der Waerden number, denoted by aw(n, k), is the least positive integer r such that every exact r-coloring of [n] contains a rainbow k-term AP.
6 aw(8, 3)
7 aw(8, 3)
8 aw(8, 3)
9 aw(8, 3)
10 aw(8, 3)
11 Properties of aw(n, k) k aw(n, k) n. aw(n, k) = n if and only if k n aw(n, k) aw(n 1, k) + 1.
12 Small anti-van der Waerden numbers n\k
13 Lowerbound for aw(n, 3) Theorem aw( n 3, 3) + 1 aw(n, 3)
14 Lowerbound for aw(n, 3) Theorem aw( n 3, 3) + 1 aw(n, 3) Corollary log 3 (n) + 2 aw(n, 3).
15 Upperbound for aw(n, 3) Theorem aw(n, 3) aw( n 2, 3) + 1.
16 Upperbound for aw(n, 3) Theorem aw(n, 3) aw( n 2, 3) + 1. Corollary aw(n, 3) log 2 (n) + 1.
17 Upperbound for aw(n, 3) Theorem aw(n, 3) aw( n 2, 3) + 1. Corollary aw(n, 3) log 2 (n) + 1. Conjecture aw(n, 3) log 3 (n) + 4. Conjecture aw(3 m, 3) m + 2.
18 aw(z n, k) A k-term arithmetic progression in Z n is a finite sequence of k terms of the form {a, a + d, a + 2d,..., a + (k 1)d}(mod n), where k, a, and d are integers.
19 Small anti-van der Waerden numbers of Z n n\k
20 Small anti-van der Waerden numbers of Z n with k =
21 Theorem aw(z 2 m, 3) = 3.
22 Theorem aw(z 2 m, 3) = 3. Theorem If t is odd and s is either odd or 2 m, then aw(z st, 3) aw(z s, 3) + aw(z t, k) 2.
23 Question: Is aw(z s, 3) + aw(z t, 3) 2 aw(z st, k) when t is odd?
24 Question: Is aw(z s, 3) + aw(z t, 3) 2 aw(z st, k) when t is odd? Question: Given a prime p, does there exist a coloring of Z p with aw(z p, 3) 1 colors, no rainbow 3-term APs, and a color class of size 1?
25 Question: Is aw(z s, 3) + aw(z t, 3) 2 aw(z st, k) when t is odd? Question: Given a prime p, does there exist a coloring of Z p with aw(z p, 3) 1 colors, no rainbow 3-term APs, and a color class of size 1? Question: Is aw(z p, 3) 4, for all prime p?
26 Thank You!
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