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.Comment: 21 page

Some counterexamples in the partition calculus

We show that the pairs (2-element subsets; edges of the complete graph) of a set of cardinality ℵ1 can be colored with 4 colors so that every uncountable subset contains pairs of every color, and that the pairs of real numbers can be colored with ℵ0 colors so that every set of reals of cardinality 2ℵ0 contains pairs of every color. These results are counterexamples to certain transfinite analogs of Ramsey's theorem. Results of this kind were obtained previously by Sierpiński and by Erdös, Hajnal, and Rado. The Erdös-Hajnal-Rado result is much stronger than ours, but they used the continuum hypothesis and we do not. As by-products, we get an uncountable tournament with no uncountable transitive subtournament, and an uncountable partially ordered set such that every uncountable subset contains an infinite antichain and a chain isomorphic to the rationals. The tournament was pointed out to us by R. Laver, and is included with his permission

Borel\u27s Conjecture in Topological Groups

We introduce a natural generalization of Borel\u27s Conjecture. For each infinite cardinal number κ, let BCκ denote this generalization. Then BCℵ0 is equivalent to the classical Borel conjecture. Assuming the classical Borel conjecture, ¬BCℵ1 is equivalent to the existence of a Kurepa tree of height ℵ1. Using the connection of BCκ with a generalization of Kurepa\u27s Hypothesis, we obtain the following consistency results: 1. If it is consistent that there is a 1-inaccessible cardinal then it is consistent that BCℵ1. 2. If it is consistent that BCℵ1, 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 ¬BCℵω+(∀n\u3cω)BCℵn is consistent. 4. If it is consistent that there is a 2-huge cardinal, then it is consistent that BCℵω. 5. If it is consistent that there is a 3-huge cardinal, then it is consistent thatBCκ for a proper class of cardinals κ of countable cofinality

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 $c(G)+c(\overline G)=n+2$.Comment: 22 page

UCC Civic Engagement Plan 2017-2022.

This civic engagement plan was drafted by the University Civic and Community Engagement Committee, following extensive consultations with staff, students and community stakeholders. It is grounded in a benchmarking exercise, a review of international literature and best practices, and is informed by a self assessment of UCC’s engagement activity in 2016, which was supported by the Carnegie Foundation. A staff survey conducted in 2016 found that there was reasonable staff activity in the area of community engagement. However, staff cited barriers such as having insufficient time, a lack of recognition or valuing of engagement, and engagement needing to be integral to the mission of the University. They further referred to a fragmented organisational approach and needing better communication and information centrally. Recently Milward-Brown surveyed a representative sample of 400 people across the Munster region on behalf of UCC. The results showed that public understanding of the societal engagement mission of the University is low compared to other factors; underscoring the importance of more intently demonstrating and communicating the value of our engagement work to the public

Making a difference: A research report on student volunteering in UCC

Higher education institutions (HEIs) in Ireland are still at an early stage in recognising student volunteering and student-led engagement activities. Not surprisingly then, research on student volunteering, and on the potential benefits for communities, HEIs and young people themselves, is limited in the Irish context. In this report, we seek to contribute to this underresearched field of educational and social research. Based on a survey of over 2,000 students at University College Cork (UCC), the report represents one of the most comprehensive studies of student volunteering in Ireland to date

Community research report

University College Cork introduced its first Community-based Participatory Research (CBPR) module in 2016. The module was funded and supported by Horizon2020 funding, specifically the EnRRICH project (Enhancing Responsible Research and Innovation through Curricula in Higher Education). The module is a 5-credit module for PhD students from all disciplines in the early stages of their PhD at University College Cork. Following two fruitful partnerships in the areas of social justice / equality, community family support services and older persons, there was a keen interested to explore partnerships in markedly different areas such as environmental sustainability. A dialogue ensued with CEF where the opportunity and feasibility to collaborate on the CBPR module was explored