Related papers: Coloring ordinals by reals

We study combinatorial principles we call Homogeneity Principle HP(\kappa) and Injectivity Principle IP(\kappa,\lambda) for regular \kappa>\aleph_1 and \lambda\leq\kappa which are formulated in terms of coloring the ordinals <\kappa by…

We present an explicit family of hypergraphs with arbitrarily large uniformity and chromatic number that admit realizations in both geometric and number-theoretic settings. As an application, we give a new proof of a theorem of Chen, Pach,…

We propose a natural theory SO axiomatizing the class of sets of ordinals in a model of ZFC set theory. Both theories possess equal logical strength. Constructibility theory in SO corresponds to a natural recursion theory on ordinals.

These notes are extracted from the lectures on forcing axioms and applications held by professor Matteo Viale at the University of Turin in the academic year 2011-2012. Our purpose is to give a brief account on forcing axioms with a special…

This work is concerned with an axiom introduced by Todorc\v{e}vi\'{c} in \cite{stevo} that constitutes a Ramsey-like statement regarding the topology of the reals. Our aim is to explain the axiom in detail, give some interesting…

Consider edge colorings of digraphs where edges $v_1 v_2$ and $v_2 v_3$ have different colors. This coloring induces a vertex coloring by sets of edge colors, in which edge $v_1 v_2$ in the graph implies that the set color of $v_1$ contains…

We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderable to prove that the following statements are equivalent to…

The Lov\'asz Local Lemma is a powerful probabilistic technique for proving the existence of combinatorial objects. It is especially useful for colouring graphs and hypergraphs with bounded maximum degree. This paper presents a general…

In arXiv:2209.04859 Andy Zucker and Chris Lambie-Hanson proved the consistency result for some coloring principle for the products of polish spaces by at most countable many colors. This principle easy implies Halpern and L\"auchli's…

Many of the theorems of real analysis, against the background of the ordered field axioms, are equivalent to Dedekind completeness, and hence can serve as completeness axioms for the reals. In the course of demonstrating this, the article…

P\'olya's enumeration theorem is concerned with counting labeled sets up to symmetry. Given a finite group acting on a finite set of labeled elements it states that the number of labeled sets up to symmetry is given by a polynomial in the…

We get a quite maximal version of the colouring property $Pr_1$ by proving $Pr_1(\lambda,\lambda,\lambda,\theta)$ when $\lambda = \partial^+, \partial > \theta$ are regular cardinals.

We prove a better coloring theorem for aleph_4 and even aleph_3. This has a general topology consequence.

Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to ${\sf WKL}_0$, ${\sf ACA}_0$ and $\Pi ^1_ 1$-${\sf CA}_0$.

We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of…

We prove two colorful Carath\'eodory theorems for strongly convex hulls, generalizing the colorful Carat\'eodory theorem for ordinary convexity by Imre B\'ar\'any, the non-colorful Carath\'eodory theorem for strongly convex hulls by the…

We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{\'n}ski…

We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets. Along the way, we establish new results in club-guessing and in the general theory of…

Here it is shown that standard set theory can be interpreted in a theory about order. The ordering here is about non-extensional flat classes, i.e. classes that are not elements of classes. So, stipulating a nearly well order over all those…

We construct a model of $\mathsf{MA_{\aleph_1}}+\mathsf{OCA}_T$ where Baumgartner's Axiom fails, settling a question of Farah. Moreover, in the same model there is an $\aleph_1$-dense set of reals which is neither reversible nor increasing,…