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Abstract  

We determine a class of triple systems such that each must occur in a triple system with uncountable chromatic number that omits
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(the unique system consisting of two triples on four vertices). This class contains all odd circuits of length ≧ 7. We also show that consistently there are two finite triple systems such that they can separately be omitted by uncountably chromatic triple systems but not both.
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Abstract  

We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Karłowicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some aXY the order-type of aX is different from that of aY. In addition to the chromatic number x(G) of these graphs we study χκ(G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.

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Hujter and Lángi defined the k-fold Borsuk number of a set S in Euclidean n-space of diameter d > 0 as the smallest cardinality of a family F of subsets of S, of diameters strictly less than d, such that every point of S belongs to at least k members of F.

We investigate whether a k-fold Borsuk covering of a set S in a finite dimensional real normed space can be extended to a completion of S. Furthermore, we determine the k-fold Borsuk number of sets in not angled normed planes, and give a partial characterization for sets in angled planes.

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strains Organic acids production (µg mL −1 ) Acetate Lactate Butyrate K4E (5.86 ± 0.04) c (16.43 ± 0.70) d (0.122 ± 0.002) g,h K7 (5.77 ± 0.05) c (15.78 ± 0.37) d (0.163 ± 0.002) h K16 (5.18 ± 0.02) b (11.46 ± 0.04) e (0.090 ± 0.001) g RD7 (4.07 ± 0.03) a

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