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{\bf Le Anh Vinh}
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{\bf Explicit Ramsey Graphs and Erd\H os Distance Problems over Finite Euclidean and Non-Euclidean Spaces}
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We study the Erd\H os distance problem over finite Euclidean and
non-Euclidean spaces. Our main tools are graphs associated to finite
Euclidean and non-Euclidean spaces that are considered in
Bannai-Shimabukuro-Tanaka (2004, 2007). These graphs are shown to be
asymptotically Ramanujan graphs. The advantage of using these graphs
is twofold. First, we can derive new lower bounds on the Erd\H os
distance problems with explicit constants. Second, we can construct
many explicit tough Ramsey graphs $R(3,k)$.



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