Let N^{+}(k)= 2^{k/2} k^{3/2} f(k) and N^{-}(k)= 2^{k/2} k^{1/2} g(k) where 1=o(f(k)) and g(k)=o(1). 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 k-term arithmetic progression approaches 0, for large k. This improves an upper bound due to Brown, who had established an analogous result for N^{+}(k)= 2^k log k f(k).
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) ≥ p2 p whenever p is prime, and Gowers [4] showed that W (k) is bounded above by a tower of finite height, i.e.,
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 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]
A family of sets F = {S 1 , S 2 , . . . , S m } is said to be almost disjoint if any two distinct elements of F have at most one element in common, i.e., if
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 c k denote the asymptotic constant such that the size of the largest family of almost disjoint k-term arithmetic progressions contained in [1, n] is c k n 2 /(2k -2). It follows from the above lemma that c k ≥ 1/k 2 . Perhaps there is an absolute constant λ such that c k ≤ λ/k 2 . Ardal, Brown and Pleasants [1] have shown that 0.476 ≤ c 3 ≤ 0.485.
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 2 /k 3 ), as shown in the previous section.
. . , B q of length s, and possibly one residual block B q+1 of length r.
. By Lemma , the elements of F 1 are almost disjoint, and s 2 /4k 3 ≤ |F 1 | ≤ s 2 /k 3 for large k.
For each arithmetic progression P ∈ F 1 , let C P denote the set of 2colorings of B 1 in which P is monochromatic. Then |C P | = 2 s-k+1 . Also,
Similarly, we can consider the blocks B 2 , B 3 , . . . , B q and the corresponding families F 2 , F 3 , . . . , F q . Let p 0 be the probability that no arithmetic progression from any of the F i is monochromatic under a 2-coloring chosen randomly and uniformly. Then
it follows that p 0 approaches 0 for large k. Thus the probability that some arithmetic progression is monochromatic approaches 1 as k → ∞.
where g(k) → 0 arbitrarily slowly as k → ∞. Then, the probability that a 2-coloring of {1, 2, . . . , N -(k)} chosen randomly and uniformly contains a monochromatic k-AP approaches 0 as k → ∞.
Proof. Let n = N -(k), and let E be the expected number of monochromatic k-APs in a 2-coloring of {1, 2, . . . , n} chosen randomly and uniformly. Note that there are n 2 (1 + o(1))/(2k -2) k-APs contained in [1, n] and each of these is monochromatic with probability 2 1-k . By linearity of expectation, E < k[g(k)] 2 /(k -2). For r ≥ 0, let p r be the probability that there are exactly r monochromatic k-APs in a random 2-coloring. Then E = p 1 + 2p 2 + 3p 3 + . . . > 1 -p 0 , so that p 0 > (k -2 -k[g(k)] 2 )/(k -2). Thus, the probability that some arithmetic progression is monochromatic approaches 0 as k → ∞.
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