Monochromatic progressions in random colorings

Let N(k) = 2k/2k3/2f(k) and N(k) = 2k/2k1/2 g(k) where f(k) → ∞ and g(k) → 0 arbitrarily slowly as k → ∞. We show that the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic kterm arithmetic progression approaches 0, as k → ∞. This improves an upper bound due to Brown [3], who had established an analogous result for N(k) = 2k log kf(k). Introduction One of the earliest results in Ramsey theory is the theorem of van der Waerden [5], stating that for any positive integer k, there exists an integer W (k) such that any 2-coloring of {1, 2, . . . ,W (k)} yields a monochromatic k-term arithemtic progression. The exact values of W (k) are known only for k ≤ 6. Berlekamp [2] showed that W (p+1) ≥ p2p whenever p is prime, and Gowers [4] showed that W (k) is bounded above by a tower of finite height, i.e., W (k) ≤ 2 2 2 2 k+9 Since the best known upper and lower bounds on W (k) are far apart, a lot of work has been done on variants of the original problem. A natural question from a probabilistic perspective is to obtain upper bounds on the slowest growing function N(k) such that the probability of a 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic k-term arithmetic progression (hereafter abbreviated as k-AP) approaches 1 as k → ∞. Similarly, one 1 could seek lower bounds on the fastest growing function N(k) such that the probability of a 2-coloring of {1, 2, . . . , N(k)} containing a monochromatic k-AP approaches 0 as k → ∞. An upper bound for N(k) was established by Brown [3] who showed that N(k) = O(2k log kf(k)) where f(k) → ∞ as k → ∞. We improve this bound to N(k) = O(2k/2k3/2f(k)), and also show that N(k) = Ω(2k/2k1/2 g(k)) where g(k) → 0 as k → ∞. Almost Disjoint Progressions A family of sets F = {S1, S2, . . . , Sm} is said to be almost disjoint if any two distinct elements of F have at most one element in common, i.e., if |Si ∩ Sj| ≤ 1 whenever i 6= j. Lemma. Let Fk,n be the collection of k-APs contained in {1, 2, . . . , n} with common difference d satisfying n/k ≤ d < n/(k−1). Then Fk,n is an almost disjoint family. Moreover, |Fk,n| = n (1 + o(1))/2k. Proof. Let n/k ≤ d < n/(k − 1), and let A1 = {a, a + d, . . . , a + (k − 1)d} be a k-term arithmetic progression in Fk,n. For 0 ≤ l ≤ k − 1, consider the pairwise disjoint half-open intervals Il = (ln/k, (l + 1)n/k]. We claim that a + ld ∈ Il for 0 ≤ l ≤ k − 1. Clearly, a + ld > ld ≥ ln/k. Moreover, a+ ld = a+ (k − 1)d − (k − l− 1)d ≤ n− (k − l− 1)n/k = (l+ 1)n/k. In particular, a ≤ n/k ≤ d. Now suppose that A2 = {a , a + d, . . . , a + (k − 1)d} ∈ Fk,n with |A1 ∩ A2| ≥ 2. Let a + l1d = a ′ + l1d ′ and a + l2d = a ′ + l2d . Since the intervals Il are pairwise disjoint, it follows that l1 = l ′ 1 and l2 = l ′ 2. But then we have a = a and d = d, so that A1 = A2. Thus Fk,n is an almost disjoint family. Finally, |Fk,n| = ∑ n k ≤d< n k−1 (n− d(k − 1)) = n(1 + o(1)) 2k , since there are n− d(k − 1) k-term arithmetic progressions of common difference d completely contained in {1, 2, . . . , n}. For each integer k ≥ 3, let ck denote the asymptotic constant such that the size of the largest family of almost disjoint k-term arithmetic progressions contained in [1, n] is ckn /(2k − 2). It follows from the above lemma 2 that ck ≥ 1/k . Perhaps there is an absolute constant λ such that ck ≤ λ/k . Ardal, Brown and Pleasants [1] have shown that 0.476 ≤ c3 ≤ 0.485. Monochromaticity: Almost Surely and Almost Never Theorem 1. Let N(k) = 2k/2k3/2f(k) where f(k) → ∞ arbitrarily slowly as k → ∞. Then the probability that a 2-coloring of {1, 2, . . . , N(k)} chosen randomly and uniformly contains a monochromatic k-term arithmetic progression approaches 1 as k → ∞. Proof. Our approach will be similar to that of Brown[3], but rather than work with a family of combinatorial lines in a suitably chosen hypercube, which is an almost disjoint family of size O(n), we work with k-APs of large common difference, which is an almost disjoint family of size Ω(n/k), as shown in the previous section. Let n = N(k) = 2k/2k3/2f(k) and q = ⌊(f(k))4/3⌋. Let s = s(k) satisfy n = qs + r, 0 ≤ r < s. We divide the interval [1, n] into q blocks B1, B2, . . . , Bq of length s, and possibly one residual block Bq+1 of length r. Let F1 = Fk,s consist of all k-APs in B1 = [1, s] with common difference d satisfying s/k ≤ d < s/(k − 1). By Lemma , the elements of F1 are almost disjoint, and s/4k ≤ |F1| ≤ s /k for large k. For each arithmetic progression P ∈ F1, let CP denote the set of 2colorings of B1 in which P is monochromatic. Then |CP | = 2 s−k+1. Also, |CP ∩ CQ| = 2 s−2k+2, since |P ∩Q| ∈ {0, 1}. By Bonferroni’s inequality,