Editorial: Plotting new courses in assessment

The articles in this issue foreground some of the tensions inherent in the use of “global” summative, norm-referenced measures of literacy on the one hand, and “local”, site and classroom specific literacy assessments on the other. At a theoretical level these tensions may seem without basis given that “global” and “local” assessments seem to serve different masters and achieve different purposes. However, in reality the wash-back effect of high stakes systemic assessment on classroom work is widely accepted. Furthermore, these tensions are palpable in countries in which the results from high-stakes, high status “global” assessments can lead to the closure of schools. Several of the articles in this issue describe how teachers in schools and universities are attempting to steer a course around and between the omnipresent impact of high stakes assessments and their influence on curricula

Forcing a sparse minor

This paper addresses the following question for a given graph $H$: what is the minimum number $f(H)$ such that every graph with average degree at least $f(H)$ contains $H$ as a minor? Due to connections with Hadwiger's Conjecture, this question has been studied in depth when $H$ is a complete graph. Kostochka and Thomason independently proved that $f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determined $f(H)$ when $H$ has a super-linear number of edges. We focus on the case when $H$ has a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that if $H$ has $t$ vertices and average degree $d$ at least some absolute constant, then $f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case when $H$ has small average degree, we prove that if $H$ has $t$ vertices and $q$ edges, then $f(H) \leq t+6.291q$ (where the coefficient of 1 in the $t$ term is best possible)

Employment Relationships in the New Economy

It is often argued that 'new economy' jobs are less likely to use traditional employment relationships, and more likely to rely on 'alternative' or 'contingent' work. When we look at new economy jobs classified on the basis of employment in high-tech industries, we do not find greater use of contingent or alternative employment relationships. However, when we classify new economy workers based on residence in high-tech cities, contingent and alternative employment relationships are more common, even after accounting for the faster employment growth in these cities. Finally, defining 'new economy' more literally to be those industries with the fastest growth yields the most striking differences, as workers in the fastest-growing industries are much more likely to be in contingent or alternative employment relationships, with a large share of this difference driven by employment in the fast-growing construction and personnel supply services industries where employment is perhaps 'intrinsically' contingent or alternative. While subject to numerous qualifications, the combined evidence gives some support to the hypothesis that the new economy may entail a possibly significant and long-lasting increase in contingent and alternative employment relationships.

Polynomial treewidth forces a large grid-like-minor

Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an $\ell\times\ell$ grid minor is exponential in $\ell$. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A \emph{grid-like-minor of order} $\ell$ in a graph $G$ is a set of paths in $G$ whose intersection graph is bipartite and contains a $K_{\ell}$-minor. For example, the rows and columns of the $\ell\times\ell$ grid are a grid-like-minor of order $\ell+1$. We prove that polynomial treewidth forces a large grid-like-minor. In particular, every graph with treewidth at least $c\ell^4\sqrt{\log\ell}$ has a grid-like-minor of order $\ell$. As an application of this result, we prove that the cartesian product $G\square K_2$ contains a $K_{\ell}$-minor whenever $G$ has treewidth at least $c\ell^4\sqrt{\log\ell}$.Comment: v2: The bound in the main result has been improved by using the Lovasz Local Lemma. v3: minor improvements, v4: final section rewritte

Reverse mathematics and infinite traceable graphs

This paper falls within the general program of investigating the proof theoretic strength (in terms of reverse mathematics) of combinatorial principals which follow from versions of Ramsey's theorem. We examine two statements in graph theory and one statement in lattice theory proved by Galvin, Rival and Sands \cite{GRS:82} using Ramsey's theorem for 4-tuples. Our main results are that the statements concerning graph theory are equivalent to Ramsey's theorem for 4-tuples over \RCA while the statement concerning lattices is provable in \RCA. Revised 12/2010. To appear in Archive for Mathematical Logi