Families and work: revisiting barriers to employment

"In recent years, considerable effort has been put into supporting parents to make the transition into work. This study was commissioned by the Department for Work and Pensions (DWP) to explore whether these incentives were helping parents to overcome the barriers known to impede their engagement in the formal labour market. The report is based on fieldwork conducted in 2009. However, the concluding chapter considers the significance of the findings in light of proposals for the introduction of the Universal Credit and other reforms of the tax and benefit systems proposed by the Coalition Government." - Page 1

Control Over Work Hours and Alternative Work Schedules

[Excerpt] Alternative work schedules encompass work hours that do not necessarily fall inside the perimeters of the traditional and often rigid 8-hour workday or 40-hour workweek. Such schedules allow working people to earn a paycheck while having the flexibility to take care of children, older relatives and other needs. Examples of such schedules include: limits on mandatory overtime, flexible work day, compressed workweek, shift swap and telecommuting. Changes in the workforce and the economy are making alternative work schedules increasingly important for working families trying to balance jobs and family responsibilities

Cross-intersecting sub-families of hereditary families

Families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ of sets are said to be \emph{cross-intersecting} if for any $i$ and $j$ in $\{1, 2, ..., k\}$ with $i \neq j$, any set in $\mathcal{A}_i$ intersects any set in $\mathcal{A}_j$. For a finite set $X$, let $2^X$ denote the \emph{power set of $X$} (the family of all subsets of $X$). A family $\mathcal{H}$ is said to be \emph{hereditary} if all subsets of any set in $\mathcal{H}$ are in $\mathcal{H}$; so $\mathcal{H}$ is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family $\mathcal{H} \neq \{\emptyset\}$ of $2^X$ and any $k \geq |X|+1$, both the sum and product of sizes of $k$ cross-intersecting sub-families $\mathcal{A}_1, \mathcal{A}_2, ..., \mathcal{A}_k$ (not necessarily distinct or non-empty) of $\mathcal{H}$ are maxima if $\mathcal{A}_1 = \mathcal{A}_2 = ... = \mathcal{A}_k = \mathcal{S}$ for some largest \emph{star $\mathcal{S}$ of $\mathcal{H}$} (a sub-family of $\mathcal{H}$ whose sets have a common element). We prove this for the case when $\mathcal{H}$ is \emph{compressed with respect to an element $x$ of $X$}, and for this purpose we establish new properties of the usual \emph{compression operation}. For the product, we actually conjecture that the configuration $\mathcal{A}_1 = \mathcal{A}_2 = ... = \mathcal{A}_k = \mathcal{S}$ is optimal for any hereditary $\mathcal{H}$ and any $k \geq 2$, and we prove this for a special case too.Comment: 13 page