Sequences of Integers Avoiding 3-term Arithmetic Progressions

The optimal length r(n) of a sequence in [1, n] containing no 3term arithmetic progression is determined for several new values of n and some results relating to the subadditivity of r are obtained. We also prove a particular case of a conjecture of Szekeres. A subsequence S = (a1, a2, . . . , ak) of the sequence 〈n〉 = (1, 2, . . . , n) containing no three terms ap, aq, and ar for which aq − ap = ar − aq (i.e., S contains no three term arithmetic progression) is called an A sequence in 〈n〉. r(n) denotes the maximum number of terms possible in an A sequence in 〈n〉, and any such sequence is said to be optimal in 〈n〉. Throughout this paper any input variable x in r(x) is assumed to be a positive integer. The following properties of A sequences and the function r are evident. (P1) If S = (a1, a2, . . . , ak) is an A sequence in 〈n〉, then (n+1−ak, n+1−ak−1, . . . , n− a1) is an A sequence called the complement of S in 〈n〉. Also, for any integer j < a1, a translate (a1 − j, a2 − j, . . . , ak − j) of S is an A sequence. (P2) For any m and n, r(m+ n) ≤ r(m) + r(n). In brief, the function r is subadditive. (P3) For any n, r(n) ≤ r(n + 1) ≤ 1 + r(n). Whenever r(n − 1) < r(n), we call n a jump node for r. (P4) If (a1, a2, . . . , ak−1, ak) is an A sequence in 〈n〉, then (a1, a2, . . . , ak−1, ak, 2n− 1+ a1, 2n − 1 + a2, . . . , 2n − 1 + ak−1, 2n − 1 + ak) is an A sequence in 〈3n − 1〉; whence r(3n− 1) ≥ 2r(n). (P5) If r(n− 1) < r(n), then any optimal A sequence in 〈n〉 contains both 1 and n. (P6) If r(n− 1) < r(n) < r(n+ 1), then any optimal A sequence in 〈n+ 1〉 contains all four of 1, 2, n, and n+ 1. Observe that, by (P6), no three consecutive integers can all be jump nodes for r. the electronic journal of combinatorics 19 (2012), #P27 1 The study of A sequences was initiated by Erdős and Turan in [1], and since the appearance of their paper there has been extensive research concerning the asymptotic behavior of the function r and its correspondent that counts the sequences in 〈n〉 avoiding k-term arithmetic progressions for k > 3. A substantial paper by Szemeredi [2] gives many references on this topic. The exact value of r(n) is, however, known for only a few n. In this regard, an error in [1] in computing r(20) has gone undetected and as a consequence, subsequent computations of r(n) for certain n > 20 are based on flawed arguments. For example, the evaluations of r(21) and r(41) (and perhaps r(22) and r(23) also) in [1] are founded on incorrect reasoning. The values of r(n) for n ≤ 19 found in [1] are, however, all correct. We summarize these values by listing only the jump nodes for r: r(2) = 2, r(4) = 3, r(5) = 4, r(9) = 5, r(11) = 6, r(13) = 7, r(14) = 8. The next jump node for r after 14 is 20 and not 21 as claimed in [1]. This is because r(19) = 8, and (1, 2, 6, 7, 9, 14, 15, 18, 20) is an A sequence. There is a sequence {Tk} of positive integers with three intriguing questions surrounding it: (a) Is each Tk, k > 1, a jump node for r? (b) Is the optimal A sequence in 〈Tk〉 for each k unique? (c) Is it true that r(Tk) = 2 k for each k? The sequence {Tk} is defined recursively as follows: Tk = 3Tk−1 − 1 for k ≥ 1; T0 = 1. Observe that Tk = 1 2 (3 + 1), and that by (P4)