Generalizing König's infinity lemma

Mapping among the nodes of infinite trees: A variation of Kőnig's infinity lemma

Koenig’s infinity lemma states that an infinite rooted tree in which every node has finite degree has an infinite path. A variation of this lemma about mappings from one tree to another is presented in this note. Its proof utilizes Koenig’s lemma, and Koenig’s lemma follows from this variation.

Realization of aperiodic subshifts and densities in groups

A Theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a 2-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet {0, 1}. In this article, we use Lovász local lemma to first give a new simple proof of this result, and second to prove the ex...

Separation and Weak König's Lemma

We continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach...

König's edge coloring theorem without augmenting paths

Comments and Corrections

The proof of Theorem 1 (stochastic Barbalat’s lemma) in the paper by Wu et al. is incorrect. This note provides a new statement of stochastic Barbalat’s lemma. In addition, some definitions, propositions, and proofs are given to replace those in the paper by Wu et al.