Explicit Ramsey Graphs and Erdős Distance Problems over Finite Euclidean and Non-Euclidean Spaces

Abstract

We study the Erdös 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 Erdös distance problems with explicit constants. Second, we can construct many explicit tough Ramsey graphs R(3, k).

The sovability of norm, bilinear and quadratic equations over finite fields via spectra of graphs

“…In [29], the author gave another proof of this result using the graph theoretic method (see also [35] for a similar proof). The (common) main step of these proofs is to estimate the number of occurrences of a fixed distance.…”

On the Sums of Any $k$ Points in Finite Fields

“…However, the exponent (d + 1)/2 has not been improved for higher even dimensions d ≥ 4. For further discussion on distance problems in finite fields, readers may refer to [5,9,16,17,18,19,26,27]. See also [3,4], and references contained therein for recent results on the distance problems in the ring setting.…”

Commutative association schemes

“…24 In fact, the same method works for many other examples of controlling association schemes; see [22,23]. See also [102,51] for related constructions of Ramanujan graphs and [174] for an application of the results in [22,23] to the Erdős distance problem. 19 These association schemes arise from the action of O 2m+1 (q) on each of the sets of plus-type and minus-type hyperplanes.…”