Graphs with many strong orientations

We establish mild conditions under which a possibly irregular, sparse graph $G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\log_2{n}$ and has Cheeger constant at least $c_2\frac{\log_2\log_2{n}}{\log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we provide an explicit construction to show our minimum degree condition is tight while the Cheeger constant bound is tight up to a $\log_2\log_2{n}$ factor.Comment: 14 pages, 4 figures; revised version includes more background and minor changes that better clarify the expositio

What will I be and how will I get there?: Examining the transition to adulthood among care leavers

Care leavers (adults formerly in foster care) are more likely to have negative outcomes in adulthood than non-fostered peers, especially in employment, earnings, and education (Courtney et al., 2011; Courtney et al., 2018; Pecora et al., 2005; Pecora et al., 2003). Success is determined by how well care leavers are able to demonstrate positive outcomes in these domains, but these domains are often defined by policy and research. Services provided by legislation focus on independent living skills to promote care leavers’ educational and employment opportunities in adulthood (Collins, 2014). However, little research has explored how care leavers themselves define success, determine their own goals, and use the services provided to meet their goals. Informed by the identity capital model (Côté, 2016b), this study answers the questions: 1) how do care leavers define success in their own words, 2) what self-defined goals did care leavers have as they transitioned out of care, and 3) what human, social, and cultural capital was available to help care leavers meet their goals at transition. Using a narrative approach, 15 care leavers were asked to offer their own definition of success, goals at transition, and provide details into what human, social, and cultural capital resources they had available to meet their goals. Findings indicate care leavers’ definitions of success demonstrate a focus on achievement, life satisfaction, and connection, and their goals are aligned with those determined by legislation and research. However, many had yet to achieve their transition goals by the time they aged out of aftercare services. This delay was based on systemic barriers that inhibited care leavers from building various capital during their time in care and during their transition to adulthood; these barriers are endemic to the child welfare system and posed a form of structural oppression in the lives of children and care leavers. This indicates a clear need for policy, practice, and research to determine better ways to provide services and reduce the impact of structural oppression within the child welfare system for future care leavers during their time in foster care, the transition from foster care, and early adulthood

Maker-Breaker Rado games for equations with radicals

We study two-player positional games where Maker and Breaker take turns to select a previously unoccupied number in $\{1,2,\ldots,n\}$. Maker wins if the numbers selected by Maker contain a solution to the equation $$x_1^{1/\ell}+\cdots+x_k^{1/\ell}=y^{1/\ell}$$ where $k$ and $\ell$ are integers with $k\geq2$ and $\ell\neq0$, and Breaker wins if they can stop Maker. Let $f(k,\ell)$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\ldots,x_k$ are not necessarily distinct, and let $f^*(k,\ell)$ be the smallest positive integer $n$ such that Maker has a winning strategy when $x_1,\ldots,x_k$ are distinct. When $\ell\geq1$, we prove that, for all $k\geq2$, $f(k,\ell)=(k+2)^\ell$ and $f^*(k,\ell)=(k^2+3)^\ell$; when $\ell\leq-1$, we prove that $f(k,\ell)=[k+\Theta(1)]^{-\ell}$ and $f^*(k,\ell)=[\exp(O(k\log k))]^{-\ell}$. Our proofs use elementary combinatorial arguments as well as results from number theory and arithmetic Ramsey theory.Comment: 18 pages, 1 figur