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We investigate some coloring properties of Kneser graphs. A star-free coloring is a proper coloring $c:V(G)\to \Bbb{N}$ such that no path with three vertices may be colored with just two consecutive numbers. The minimum positive integer $t$ for which there exists a star-free coloring $c: V(G) \to \{1,2,..., t\}$ is called the star-free chromatic number of $G$ and denoted by $\chi_s(G)$. In view of Tucker-Ky Fan's lemma, we show that for any Kneser graph ${\rm KG}(n,k)$ we have $\chi_s({\rm KG}(n,k))\geq \max\{2\chi({\rm KG}(n,k))-10, \chi({\rm KG}(n,k))\}$ where $n\geq 2k \geq 4$. Moreover, we show that $\chi_s({\rm KG}(n,k))=2\chi({\rm KG}(n,k))-2=2n-4k+2$ provided that $n \leq {8\over 3}k$. This gives a partial answer to a conjecture of [12]. Also, we conjecture that for any positive integers $n\geq 2k \geq 4$ we have $\chi_s({\rm ...
Using ultrafilter techniques we show that in any partition of $\mathbb{N}$ into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind $a,b,a^b$ (with $a,b>1$). Also, we will show that any multiplicative $IP^*$ set is an "exponential $IP$ set", the analogue of an $IP$ set with respect to exponentiation.
Comment: This paper has been withdrawn due to an error in Lemma 8
Comment: Four axioms replaced by two; two references added; Fig.6 corrected
The paper that was here is a preprint that was never turned into a proper paper. In particular it does not have enough citations to the literature. The paper "The Slice Algorithm For Irreducible Decomposition of Monomial Ideals" contains a much better description of the Label algorithm than this preprint did. If you still wish to read the original preprint then access the arXiv's version 1 of this paper, instead of version 2 which is what you are reading now.
We extend the definition of the Costas property to functions in the continuum, namely on intervals of the reals or the rationals, and argue that such functions can be used in the same applications as discrete Costas arrays. We construct Costas bijections in the real continuum within the class of piecewise continuously differentiable functions, but our attempts to construct a fractal-like Costas bijection there are successful only under slight but necessary deviations from the usual arithmetic laws. Furthermore, we are able, contingent on the validity of Artin's conjecture, to set up a limiting process according to which sequences of Welch Costas arrays converge to smooth Costas bijections over the reals. The situation over the rationals is different: there, we propose an algorithm of great generality and flexibility for the construct...
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