Cycles are strongly Ramsey-unsaturated

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp who also proved that cycles (except for $C_4$) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle $C_n$, unless n is even and adding the chord creates an odd cycle. We prove this conjecture for large cycles by showing a stronger statement: If a graph H is obtained by adding a linear number of chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum degree of H is bounded, H is either bipartite (for even n) or almost bipartite (for odd n), and n is large. This motivates us to call cycles strongly Ramsey-unsaturated. Our proof uses the regularity method

You don't know what's around the corner: A qualitative study of professional footballers in England facing career-transition

Career transition in sport is a rapidly growing area within the field of sport psychology. Interest in this area has been fuelled by the need for an increased number of professional athletes seeking support and assistance during transition from sport. However, whilst research in this field has focused in on a wide range of sports, specific research on retirement in professional football has been limited. Because of this it is argued that current research may fail to consider specific issues associated with the transition from professional football. Therefore, in an attempt to add to the existing body of research the current study aimed to provide an in-depth insight into how professional footballers understand their ‘lived-world’ during exit from their sport. A total of eight former professional footballers, who were at the time experiencing the possibility of career-transition, were interviewed in two separate focus group discussions. The interviews were analysed using interpretive phenomenological analysis (IPA). The key findings from the research show that a lack of control over their lives, lack of pre-planning and preparation for retirement as well as support and ability to seek it led professional footballers to experience heightened levels of anxiety, uncertainty and fear for their futures as well as an unexpected sense of rejection during career transition. These findings have implications for support organisations and those interested in the life-long welfare of professional footballers. It is proposed that an emphasis on pre-planning and preparation, provisions of support and encouraging help-seeking may aid professional footballers during the process out of their sport

Filter cassette for high volume air sampler

Filter cassette eliminates or substantially reduces contamination of filter media by extraneous material and facilitates handling

A Tribute to D. Dortch Warriner

I first met David Dortch Warriner in the spring of 1976 when I called him at his Emporia, Virginia office to see if he had a position available in his law firm for a lawyer just out of law school. Although I had not met him prior to my call, I knew a little of him because he was a district chairman in the Republican Party of Virginia at the time

A visual sense of number

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Use of Whatman-41 filters in air quality sampling networks (with applications to elemental analysis)

The operation of a 16-site parallel high volume air sampling network with glass fiber filters on one unit and Whatman-41 filters on the other is reported. The network data and data from several other experiments indicate that (1) Sampler-to-sampler and filter-to-filter variabilities are small; (2) hygroscopic affinity of Whatman-41 filters need not introduce errors; and (3) suspended particulate samples from glass fiber filters averaged slightly, but not statistically significantly, higher than from Whatman-41-filters. The results obtained demonstrate the practicability of Whatman-41 filters for air quality monitoring and elemental analysis