Talk:Kobon triangle problem

Fujimura Kobon

• Fujimura Kobon (03 May 1903—19 February 1983)

On the Kobon triangle problem for pseudolines

I feel it would be appropriate to add a large section for pseudoline arrangements, considering that there has been a large amount of work make specifically in that aspect of the problem, both for its own sake and for the sake of its usefulness toward the line arrangement version. Many sources already used here have an entirely separate table and section specifically for pseudolines for these reasons. BagLuke (talk) 23:08, 23 February 2025 (UTC)

Predictions...

Anyone else seeing a pattern emerging?

Boldface numbers are already verified as of August 3, 2025-- the rest are predictions.

Let me know if you're interested in the source for the calculation...

k	N(k)	U
22	145	147
23	161	161
24	173	175
25	191	191
26	207	207
27	225	225
28	239	244
29	261	261
30	278	278
31	299	299
32	318	320
33	341	341
34	364	364
35	383	383
36	406	406
37	429	429
38	456	456
39	480	480
40	506	506
41	532	532
42	562	562
43	588	588
44	616	616
45	644	644
46	674	674
47	702	702
48	736	736
49	770	770
50	806	806
51	836	836
52	869	869
53	902	902
54	939	939
55	974	974
56	1011	1011
57	1048	1048
58	1089	1089
59	1126	1126
60	1167	1167
61	1208	1208
62	1251	1251
63	1290	1290
64	1331	1331
65	1372	1372
66	1419	1419
67	1461	1461
68	1507	1507
69	1553	1553
70	1601	1601
71	1647	1647
72	1697	1697
73	1747	1747
74	1799	1799
75	1845	1845
76	1893	1893
77	1941	1941

JReimerMorales (talk) 04:27, 4 August 2025 (UTC)

Clarification on upper bounds

The paper from Bartholdi, Nicolas; Blanc, Jérémy; Loisel, Sébastien (2008) has the upper bounds for even k:

$\lfloor k(k-7/3)/3\rfloor$

However, the wiki has this as a bounds for both simple and non-simple arrangements, while the paper seems to only be focused on simple arrangements, with the bounds seemingly only applying to them. Is there some source clarifying that the bounds also apply for non-simple arrangements? David Eppstein, mind taking a look? Thanks, BagLuke (talk) 17:35, 27 April 2026 (UTC)

You appear to be correct: the source is only for simple arrangements. The optimal 6-line solution is not simple, so considering only simple arrangements appears to be a nontrivial additional assumption. —David Eppstein (talk) 18:17, 27 April 2026 (UTC)

6-line can be achieved by simple or non-simple (see the blue triangle image, alter by making parallel lines connect arbitrarily on the outside), but the 8-line and 12-line ones appear to only be possible via non-simple arrangements, so this still applies. BagLuke (talk) 18:22, 27 April 2026 (UTC)