Embedding into bipartite graphs

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and sublinear bandwidth appears as a subgraph of any $2n$-vertex graph $G$ with minimum degree $(1+\gamma)n$, provided that $n$ is sufficiently large. We show that this threshold can be cut in half to an essentially best-possible minimum degree of $(\frac12+\gamma)n$ when we have the additional structural information of the host graph $G$ being balanced bipartite. This complements results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding minimum degree threshold for $K_{r,s}$-factors, with $r$ and $s$ fixed. Moreover, it implies that the set of Hamilton cycles of $G$ is a generating system for its cycle space.Comment: 16 pages, 2 figure

19th century real analysis, forward and backward

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.Comment: 28 pages, to appear in Antiquitates Mathematica