A341822 Length of the longest 2-increasing sequence of positive integer triples with entries <= n.

1, 2, 4, 8, 10, 14, 17, 21, 27, 30, 35

Offset: 1

Author

Marcel K. Goh, Feb 20 2021

Keywords

• nonn
• hard
• more

Comments

A triple t=(a_1,a_2,a_3) is defined to be 2-less than a triple u=(b_1,b_2,b_3) if a_i < b_i for at least two coordinates i. A sequence t^(j) of triples is 2-increasing if for all i < j, t^(i) is 2-less than t^(j).

Terms n <= 5 have been confirmed by brute-force search (Table 1 of Gowers and Long (2021)).

Examples

			For n=4, the sequence (1,1,1), (1,2,2), (2,1,3), (2,2,4), (3,3,1), (3,4,2), (4,3,3), (4,4,4) has length a(4)=8 and every 2-increasing sequence of length 9 must contain a triple with some coordinate equal to 5.
		

References

• W. T. Gowers and J. Long, The length of an s-increasing sequence of r-tuples, Combinatorics, Probability and Computing 30 (2021), 1-36.

Links

• W. T. Gowers and J. Long, The length of an s-increasing sequence of r-tuples, arXiv:1609.08688 [math.CO], 2016.
• Po-Shen Loh, Directed paths: from Ramsey to Ruzsa and Szemerédi, arXiv:1505.07312 [math.CO], 2015.

Crossrefs

Cf. A000093.

Formula

a(n) >= n^{3/2} when n is a perfect square.

It is conjectured that a(n) <= n^{3/2} for all n.

Extensions

Edited by N. J. A. Sloane, Mar 21 2021

a(10)-a(11) and confirmation of previous terms by Bert Dobbelaere, Mar 27 2021