A298755 Quantitative (discrete) Helly numbers for the integer lattice Z^2.

4, 6, 6, 6, 8, 7, 8, 9, 8, 8, 10, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 11, 11, 12, 12, 12, 13, 12, 12, 13, 13, 13, 13, 14, 14, 13, 13, 14, 14, 14, 14, 14, 14, 14, 15, 14, 15, 15, 15, 15, 15, 16, 15, 16, 15, 16, 16, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 16, 17, 17, 17, 17, 17

Offset: 0

Author

Bernardo González Merino, Jan 26 2018

Keywords

• nonn

Comments

a(n) = c(Z^2,n) is the smallest k>0 such that for every collection of convex sets C_1, ..., C_m having n points of Z^2 in common, there exists a subset of this collection of at most k elements such that they still contain exactly n points of Z^2 in common.

c(Z^2,n) = g(Z^2,n) = A298562(n) for n = 0, 1, ..., 200, but it is not known whether they agree for every n or not.

Links

• G. Averkov, B. González Merino, I. Paschke, M. Schymura, and S. Weltge, Tight bounds on discrete quantitative Helly numbers, arXiv:1602.07839 [math.CO], 2016. See Fig. 3 p. 5.
• G. Averkov, B. González Merino, I. Paschke, M. Schymura, and S. Weltge, Tight bounds on discrete quantitative Helly numbers, Adv. in Appl. Math., 89 (2017), 76--101.

Crossrefs

Cf. A298562.

Formula

a(n) = max_{m=0..n} (A298562(m) + m - n). [Averkov et al.] - Andrey Zabolotskiy, Oct 02 2023

Extensions

a(31) onwards from Andrey Zabolotskiy, Oct 02 2023