| Title:
|
A De Bruijn-Erdős theorem for $1$-$2$ metric spaces (English) |
| Author:
|
Chvátal, Vašek |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
45-51 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A special case of a combinatorial theorem of De Bruijn and Erdős asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals $1$ or $2$. (English) |
| Keyword:
|
line in metric space |
| Keyword:
|
De Bruijn-Erd\H os theorem |
| MSC:
|
05D99 |
| MSC:
|
51G99 |
| idZBL:
|
Zbl 06391474 |
| idMR:
|
MR3247442 |
| DOI:
|
10.1007/s10587-014-0081-1 |
| . |
| Date available:
|
2014-09-29T09:32:03Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143947 |
| . |
| Reference:
|
[1] Aboulker, P., Bondy, A., Chen, X., Chiniforooshan, E., Miao, P.: Number of lines in hypergraphs.Discrete Appl. Math. 171 (2014), 137-140. Zbl 1288.05185, MR 3190588, 10.1016/j.dam.2014.02.008 |
| Reference:
|
[2] Chen, X., Chvátal, V.: Problems related to a De Bruijn-Erdős theorem.Discrete Appl. Math. 156 (2008), 2101-2108. Zbl 1157.05019, MR 2437004, 10.1016/j.dam.2007.05.036 |
| Reference:
|
[3] Chiniforooshan, E., Chvátal, V.: A De Bruijn-Erdős theorem and metric spaces.Discrete Math. Theor. Comput. Sci. 13 (2011), 67-74. Zbl 1283.52022, MR 2812604 |
| Reference:
|
[4] Bruijn, N. G. De, Erdős, P.: On a combinatorial problem.Proc. Akad. Wet. Amsterdam 51 (1948), 1277-1279. Zbl 0032.24405, MR 0028289 |
| Reference:
|
[5] Erdős, P.: Three point collinearity, Problem 4065.Am. Math. Mon. 50 (1943), 65; Solutions in vol. 51 (1944), 169-171. MR 1525919 |
| . |
