INFINITE COMBINATORICS PLAIN AND SIMPLE

We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

Jeux transgressifs

In the textualized universe of Denis Roche, the poetic impulse is inextricably linked to the unstable framework in which such matter is cast. Utterances of language translate the power of transgression.The text is birthed as an object of inquiry, ultimately to be dismantled : the poem is both “inadmissible” and “inexistant.” As a self-fixed interrogator focused upon the residual matter that it is and which it only can be, it is no longer subservient to the references it struggles to invoke, much less to signify. Writing, then, is an unending auto-reflexive process of annihilation and reification, and subsists, as such, to the sole extent that it disassociates from all conventional voices of articulation. More obtusely, to poeticize is to re-transcribe — otherly and with abruptness — the withered sputtering(s) of the artistic imperative. Poetry, like photography, aims and frames, slices the world into sequences and images — partial, scattered, at once undone so as to be resuscitated, briefly, again. The extant is eschewed. At stake and at center-stage : a propelled form of motivity, speed, accelerations and dead-stops, an aleatory array of unremitting shifts, “instantanées” — obstinately un-emblematic. Raging or controlled, the poem or the photo can only be a non-representational sliver, the transitory residue of an infinite combinatorics of possibles. Hence, arbitrary breaches, clamoring interruptions, ludic contortions of incompleteness. A curious dynamic.

Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

We define combinatorial principles which unify and extend the classical results of Steinhaus and Piccard on the existence of interior points in the distance set. Thus the measure and category versions are derived from one topological theorem on interior points applied to the usual topology and the density topology on the line. Likewise we unify the subgroup theorem by reference to a Ramsey property. A combinatorial form of Ostrowski's theorem (that a bounded additive function is linear) permits the deduction of both the measure and category automatic continuity theorems for additive functions.

Infinite combinatorics in function spaces: Category methods

The infinite combinatorics here give statements in which, from some sequence, an infinite subsequence will satisfy some condition - for example, belong to some specified set. Our results give such statements generically - that is, for 'nearly all' points, or as we shall say, for quasi all points - all off a null set in the measure case, or all off a meagre set in the category case. The prototypical result here goes back to Kestelman in 1947 and to Borwein and Ditor in the measure case, and can be extended to the category case also. Our main result is what we call the Category Embedding Theorem, which contains the Kestelman-Borwein-Ditor Theorem as a special case. Our main contribution is to obtain function wise rather than point wise versions of such results. We thus subsume results in a number of recent and related areas, concerning e.g., additive, subadditive, convex and regularly varying functions.