Geodesic
On the Realization of Subgraphs of a Random Graph by Diameter Graphs in Euclidean Spaces
Matematičeskie zametki, Tome 97 (2015) no. 5, pp. 699-717

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The present paper is motivated by Borsuk's classical problem of the partition of sets in spaces into parts of smaller diameter. We obtain sharp estimates for the maximal number of vertices of the induced subgraph of a random graph that, with high probability, is isomorphic to the diameter graph with given chromatic number in a space of any fixed dimension.
Keywords: random graph, diameter graph, Euclidean space, Borsuk problem, chromatic number, independence number. 
@article{MZM_2015_97_5_a5,
     author = {A. A. Kokotkin and A. M. Raigorodskii},
     title = {On the {Realization} of {Subgraphs} of a {Random} {Graph} by {Diameter} {Graphs} in {Euclidean} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {699--717},
     publisher = {mathdoc},
     volume = {97},
     number = {5},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_97_5_a5/}
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A. A. Kokotkin; A. M. Raigorodskii. On the Realization of Subgraphs of a Random Graph by Diameter Graphs in Euclidean Spaces. Matematičeskie zametki, Tome 97 (2015) no. 5, pp. 699-717. http://geodesic.mathdoc.fr/item/MZM_2015_97_5_a5/