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

0,1

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

Table of n, a(n) for n=0..75.

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.

Formula

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

Crossrefs

Cf. A298562.

Sequence in context: A141328 A322345 A298562 * A035551 A087573 A201235

Adjacent sequences: A298752 A298753 A298754 * A298756 A298757 A298758

Keyword

nonn

Author

Bernardo González Merino, Jan 26 2018

Extensions

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

Status

approved