Constructions of Optical Orthogonal Codes from Finite Geometry
The link between finite geometry and various classes of error-correcting codes is well known. Arcs in projective spaces, for instance, have a close tie to linear MDS codes as well as the high-perform...
The link between finite geometry and various classes of error-correcting codes is well known. Arcs in projective spaces, for instance, have a close tie to linear MDS codes as well as the high-perform...
On Rota's Basis Conjecture
Rota conjectured that if $(B_1,\ldots,B_n)$ are disjoint bases in a rank-$n$ matroid $M$, then there are $n$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases of $M$. We prove the weaker res...
Rota conjectured that if $(B_1,\ldots,B_n)$ are disjoint bases in a rank-$n$ matroid $M$, then there are $n$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases of $M$. We prove the weaker res...
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Avoiding Monochromatic Sequences With Special Gaps
SIAM J. Discrete Math. Volume 21, Issue 3, pp. 794-801 (2007)
Published September 28, 2007
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For $S \subseteq \mathbb{Z}^+$ and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,\dots,n\}$ there must be a monochromatic sequence $\{x_{1},x_{2},\dots,x_{k}\}$ with $x_{i}-x_{i-1} \in S$ for $2 \leq i \leq k$. We consider the existence of $f(S,k;r)$ for various choices of $S$, as well as upper and lower bounds on this function. In particular, we show that this function exists for all $k$ if $S$ is an odd translate of the set of primes and $r=2$.
©2007 Society for Industrial and Applied Mathematics| History: | Received February 3, 2003; accepted May 10, 2007; published September 28, 2007 |
| Permalink: | http://dx.doi.org/10.1137/S0895480103422196 |
KEYWORDS and AMS
05D10, 11B25, 11N13
PUBLICATION DATA
0895-4801 (print)
1095-7146 (online)
SIAM