Rules and Reals

Martin Goldstern; Menachem Kojman

Home/Archive/arXiv

arXiv · math/9707204

Rules and Reals

Martin Goldstern·Menachem Kojman

View original source ↗Read source document ↗Export metadata ↓

Abstract

A ``k-rule" is a sequence A=((A_n,B_n):n<omega) of pairwise disjoint sets B_n, each of cardinality at most k, where A_n is a subset of B_n. A set X of natural numbers (a ``real'') follows a rule A if for infinitely many n we have that the intersection of X with B_n is exactly A_n. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s_k, and the least number of k-rules without a real following all of them, r_k. We investigate these cardinal invariants and their connection to some well-known cardinals from Cichon's diagram. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over omega. The consistency of such a family is still open.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Martin Goldstern, Menachem Kojman. 1997-07-16. Rules and Reals. https://arxiv.org/abs/math/9707204

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ultrahomogeneous hypertournaments

A $k$-hypertournament is a higher-arity generalisation of a tournament, consisting of a set together with an irreflexive $k$-ary relation such that the automorphism group of each substructure of size $k$ is the alternating group $\mathrm{A}_k$. We show that, for each $k \geq 3$, there are continuum-many non-isomorphic countable ultrahomogeneous $k$-hypertournaments, answering a question of Cherlin.

math.LO↗

arXiv2026

Forcing with Symmetric Systems of Models of Two Types

The purpose of this paper is to present a general method for forcing on $ω_2$ and $ω_3$ with finite conditions, while preserving all cardinals and some fragments of $\mathrm{GCH}$. This method is based on the technique of forcing with finite symmetric systems of elementary submodels, and improves earlier versions of this forcing by including models of two types. We will present several applications of the pure side condition forcing and variants thereof, by adding a Kurepa tree on $ω_2$, a club subset of $ω_2$ that avoids infinite sets from the ground model, a function bounding every canonical function below $ω_3$ on a club, and a simplified $(ω_2,1)$-morass.

math.LO↗

arXiv2026

Well begun is half done: the computation of Eulerian paths

We characterize the uniform computational strength of finding (one-way or two-way, directed or undirected) Eulerian paths in countable graphs. In the one-way case, for both directed and undirected graphs, this problem is Weihrauch-equivalent to $\mathsf{lim}\times\mathsf{C}_{\mathbb{N}}'$, where $\mathsf{C}_{\mathbb{N}}'$ is the jump of closed choice on the natural numbers. This refines the proof-theoretic strength of the existence of a one-way Eulerian path given by the reverse mathematics framework. The $\mathsf{C}_{\mathbb{N}}'$ component comes entirely from the difficulty of choosing a starting vertex: with a valid start prescribed, the problem is equivalent to $\mathsf{lim}$, which also characterizes the degree of the two-way directed/undirected Eulerian path problem. We also study the complexity of orienting the edges of a graph while preserving the existence of a one-way/two-way Eulerian path. In the one-way case, this problem is not computable, and we prove that $\mathsf{lim}_2<_{\mathrm{W}}\mathsf{OrientEP}<_{\mathrm{W}}\mathsf{C}_{2}'\sqcap\mathsf{Sort}_2$ and $\mathsf{OrientEP}\not\le_{\mathrm{W}}\mathsf{C}_{\mathbb{N}}$. On the other hand, valid orientations are uniformly computable for two-way paths and for one-way paths with a prescribed valid starting vertex.

math.LO↗