Geodesic
Low rank co-diagonal matrices and Ramsey graphs
The electronic journal of combinatorics, Tome 7 (2000)
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We examine $n\times n$ matrices over $Z_m$, with 0's in the diagonal and nonzeros elsewhere. If $m$ is a prime, then such matrices have large rank (i.e., $n^{1/(p-1)}-O(1)$ ). If $m$ is a non-prime-power integer, then we show that their rank can be much smaller. For $m=6$ we construct a matrix of rank $\exp(c\sqrt{\log n\log \log n})$. We also show, that explicit constructions of such low rank matrices imply explicit constructions of Ramsey graphs.
DOI : 10.37236/1493
Classification : 05C50, 15B33
Mots-clés : composite modules, explicit Ramsey-graph constructions, matrices over rings, co-diagonal matrices 
@article{10_37236_1493,
     author = {Vince Grolmusz},
     title = {Low rank co-diagonal matrices and {Ramsey} graphs},
     journal = {The electronic journal of combinatorics},
     year = {2000},
     volume = {7},
     doi = {10.37236/1493},
     zbl = {0939.05060},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1493/}
}
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Vince Grolmusz. Low rank co-diagonal matrices and Ramsey graphs. The electronic journal of combinatorics, Tome 7 (2000). doi: 10.37236/1493

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