arXiv subjects

Fred Galvin

Publications and source records attributed to Fred Galvin.

3 recordsLinked to original sources

arXiv2023

Factoring complete graphs and hypergraphs into factors with few maximal cliques

For integers $r,t\geq2$ and $n\geq1$ let $f_r(t,n)$ be the minimum, over all factorizations of the complete $r$-uniform hypergraph of order $n$ into $t$ factors $H_1,\dots,H_t$, of $\sum_{i=1}^tc(H_i)$ where $c(H_i)$ is the number of maximal cliques in $H_i$. It is known that $f_2(2,n)=n+1$; in fact, if $G$ is a graph of order $n$, then $c(G)+c(\overline G)\geq n+1$ with equality iff $\omega(G)+\alpha(G)=n+1$ where $\omega$ is the clique number and $\alpha$ the independence number. In this paper we investigate $f_r(t,n)$ when $r>2$ or $t>2$. We also characterize graphs $G$ of order $n$ with $c(G)+c(\overline G)=n+2$.

math.CO↗

arXiv2014

Baire spaces and infinite games

It is well known that if the nonempty player of the Banach-Mazur game has a winning strategy on a space, then that space is Baire in all powers even in the box topology. The converse of this implication may be true also: We know of no consistency result to the contrary. In this paper we establish the consistency of the converse relative to the consistency of the existence of a proper class of measurable cardinals.

math.LO↗

arXiv2012

Borel's Conjecture in Topological Groups

We introduce a natural generalization of Borel's Conjecture. For each infinite cardinal number $κ$, let {\sf BC}$_κ$ denote this generalization. Then ${\sf BC}_{\aleph_0}$ is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, $\neg{\sf BC}_{\aleph_1}$ is equivalent to the existence of a Kurepa tree of height $\aleph_1$. Using the connection of ${\sf BC}_κ$ with a generalization of Kurepa's Hypothesis, we obtain the following consistency results: (1)If it is consistent that there is a 1-inaccessible cardinal then it is consistent that ${\sf BC}_{\aleph_1}$. (2)If it is consistent that ${\sf BC}_{\aleph_1}$ holds, then it is consistent that there is an inaccessible cardinal. (3)If it is consistent that there is a 1-inaccessible cardinal with $ω$ inaccessible cardinals above it, then $\neg{\sf BC}_{\aleph_ω} \, +\, (\forall n<ω){\sf BC}_{\aleph_n}$ is consistent. (4)If it is consistent that there is a 2-huge cardinal, then it is consistent that ${\sf BC}_{\aleph_ω}$. (5)If it is consistent that there is a 3-huge cardinal, then it is consistent that ${\sf BC}_κ$ holds for a proper class of cardinals $κ$ of countable cofinality.

math.LO↗