arXiv subjects

Trevor M. Wilson

Publications and source records attributed to Trevor M. Wilson.

9 recordsLinked to original sources

arXiv2023

The infinite Fibonacci cube and its generalizations

The Fibonacci cube $Γ_n$ is is the graph whose vertices are independent subsets of the path graph of length $n$, where two such vertices are considered adjacent if they differ by the addition or removal of a single element. Klavžar [1] suggested considering the infinite Fibonacci cube $Γ_\infty$ whose vertices are independent subsets of the one-way infinite path graph with the same adjacency condition. We show that every connected component of $Γ_\infty$ is asymmetric (has no nontrivial automorphism) and no two connected components of $Γ_\infty$ are isomorphic. This follows from our results on a further generalization $Γ_G$ where $G$ is a simple, locally finite hypergraph with no isolated vertices.

math.CO↗

arXiv2022

The distinguishing index of graphs with infinite minimum degree

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $α$-regular graph for some infinite cardinal $α$ then $D'(G) \le 2$, proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if $G$ is a graph with infinite minimum degree and at most $2^α$ vertices of degree $α$ for every infinite cardinal $α$, then $D'(G) \le 3$. In particular, $D'(G) \le 3$ if $G$ has infinite minimum degree and order at most $2^{\aleph_0}$.

math.CO↗

arXiv2020

The large cardinal strength of Weak Vopěnka's Principle

We show that Weak Vopěnka's Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class C there is a C-strong cardinal. Weak Vopěnka's Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing C-strongness of some cardinal.

math.LO↗

arXiv2019

Weak Vopěnka's Principle does not imply Vopěnka's Principle

Vopěnka's Principle says that the category of graphs has no large discrete full subcategory, or equivalently that the category of ordinals cannot be fully embedded into it. Weak Vopěnka's Principle is the dual statement, which says that the opposite category of ordinals cannot be fully embedded into the category of graphs. It was introduced in 1988 by Adámek, Rosický, and Trnková, who showed that it follows from Vopěnka's Principle and asked whether the two statements are equivalent. We show that they are not. However, we show that Weak Vopěnka's Principle is equivalent to the generalization of itself known as Semi-Weak Vopěnka's Principle.

math.CT↗

arXiv2019

A game-theoretic proof of Shelah's theorem on labeled trees

We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality $κ$ of the family is much larger (in the sense of large cardinals) than the cardinality $λ$ of the set of labels, more precisely if the partition relation $κ\to (ω)^{\mathord{<}ω}_λ$ holds, then there is a homomorphism from one labeled tree in the family to another. Our proof uses a characterization of such homomorphisms in terms of games.

math.LO↗

arXiv2018

On forcing projective generic absoluteness from strong cardinals

W.H. Woodin showed that if $κ_1 < \cdots < κ_n$ are strong cardinals then two-step ${\bfΣ}^1_{n+3}$ generic absoluteness holds after collapsing $2^{2^{κ_n}}$ to be countable. We show that this number can be reduced to $2^{κ_n}$, and to $κ_n^+$ in the case $n = 1$, but cannot be further reduced to $κ_n$.

math.LO↗

arXiv2018

The consistency strength of the perfect set property for universally Baire sets of reals

We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals but are much weaker: if $0^\sharp$ exists then every Silver indiscernible is virtually Shelah in $L$. We also show that the statement $\text{uB} = {\bfΔ}^1_2$, where $\text{uB}$ is the pointclass of all universally Baire sets of reals, is equiconsistent modulo ZFC with the existence of a $Σ_2$-reflecting virtually Shelah cardinal.

math.LO↗

arXiv2018

Generic Vopěnka cardinals and models of ZF with few $\aleph_1$-Suslin sets

We define a generic Vopěnka cardinal to be an inaccessible cardinal $κ$ such that for every first-order language $\mathcal{L}$ of cardinality less than $κ$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if $|\mathscr{B}| = κ$ and every structure in $\mathscr{B}$ has cardinality less than $κ$, then an elementary embedding between two structures in $\mathscr{B}$ exists in some generic extension of $V$. We investigate connections between generic Vopěnka cardinals in models of ZFC and the number and complexity of $\aleph_1$-Suslin sets of reals in models of ZF. In particular, we show that ZFC + (there is a generic Vopěnka cardinal) is equiconsistent with ZF + $(2^{\aleph_1} \not\leq |S_{\aleph_1}|)$ where $S_{\aleph_1}$ is the pointclass of all $\aleph_1$-Suslin sets of reals, and also with ZF + $(S_{\aleph_1} = {\bfΣ}^1_2)$ + $(Θ= \aleph_2)$ where $Θ$ is the least ordinal that is not a surjective image of the reals.

math.LO↗

arXiv2018

Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle

We consider a weak version of Schindler's remarkable cardinals that may fail to be $Σ_2$-reflecting. We show that the $Σ_2$-reflecting weakly remarkable cardinals are exactly the remarkable cardinals, and we show that the existence of a non-$Σ_2$-reflecting weakly remarkable cardinal has higher consistency strength: it is equiconsistent with the existence of an $ω$-Erdős cardinal. We give an application involving gVP, the generic Vopěnka principle defined by Bagaria, Gitman, and Schindler. Namely, we show that gVP + "Ord is not $Δ_2$-Mahlo" and $\text{gVP}({\bfΠ}_1)$ + "there is no proper class of remarkable cardinals" are both equiconsistent with the existence of a proper class of $ω$-Erdős cardinals, extending results of Bagaria, Gitman, Hamkins, and Schindler.

math.LO↗