# Talk:Keller's conjecture

Is it still open? Somebody posted the following links to Russian page on open problems, claiming that the conjecture was eventually proved in dimension 7 as well:

Debroni J. A complete resolution of the Keller maximum clique problem / J. Debroni, J. D. Eblen, M. A. Langston, W. Myrvold, P. Shor, D. Weerapurage // Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms (San Francisco, California, USA, January 23–25, 2011) / Edited by D. Randall. http://www.siam.org/proceedings/soda/2011/SODA11_011_debronij.pdf
Debroni J. The End to the Keller Conjecture / J. Debroni, J. D. Eblen, M. A. Langston, W. Myrvold, P. Shor, D. Weerapurage // Forty-First Southeastern International Conference on Combinatorics, Graph Theory, and Computing (March 8-12, 2010). http://math.fau.edu/cgtc/cgtc41/Abstracts/WMyrvold.pdf
Maxal (talk) 23:53, 17 April 2013 (UTC)

Our article states "Although this graph-theoretic version of the conjecture is now resolved for all dimensions, Keller's original cube-tiling conjecture remains open in dimension 7." The links you give are on the graph-theoretic version and do not resolve the question about 7d cube tilings. The issue is the following: if the graph theoretic version is false in a certain dimension d, then the tiling version is also false in d. And if the cube tiling version is false in a certain dimension d, then the graph version is false in some higher dimension d' > d. So our knowledge about the graph version doesn't allow us to determine the exact dimension at which the tiling version fails. There's a note about this in the article, in the third paragraph of the "Keller graphs" section, but perhaps it's too cryptic. —David Eppstein (talk) 00:23, 18 April 2013 (UTC)
I see. Thank you for clarification and sorry for confusion. Maxal (talk) 00:55, 18 April 2013 (UTC)

@Dedhert.Jr Currently it mentions the 1992 breakthrough in the lede, but not the 2019 solution that finished proving it for all the dimensions, just 'it is now known'. That should be done better, but you reverted my edit to do this. PhotographyEdits (talk) 07:37, 3 June 2024 (UTC)

The lead should explain the summary of the whole article. I think it's already explained in the second paragraph, stating that Lagarias and Shor in 1992 explained that conjecture on Keller's graph dimension is false in more than ten dimensions, after which the refinement version says it is false in more than seven dimensions. Your writing about the 2019 solution that finished proving it for all the dimensions (maybe did you mean at least seven dimensions?) already explained in such. Dedhert.Jr (talk) 10:55, 3 June 2024 (UTC)