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A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors to each vertex guarantees an $L$-coloring. Given a list assignment $L$, an $L$-request for a vertex $v$ is a color $c\in L(v)$. In this paper, we look at a variant of the widely studied class of precoloring extension problems from [Z. Dvo\v{r}\'ak, S. Norin, and L. Postle: List coloring with requests. J. Graph Theory 2019], wherein one must satisfy "enough", as opposed to all, of the requested set of precolors. A graph $G$ is $\varepsilon$-flexible for list size $k$ if for any $k$-list assignment $L$, and any set $S$ of $L$-requests, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. It is conjectured that planar graphs are $\varepsilon$-flexible for list size $5$, yet it is proved only for list size $6$ and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masa\v{r}\'ik and show that planar graphs without $K_4^-$ are $\varepsilon$-flexible for list size $5$. We also prove that planar graphs without $4$-cycles and $3$-cycle distance at least 2 are $\varepsilon$-flexible for list size $4$. Finally, we introduce a new (slightly weaker) form of $\varepsilon$-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are $\varepsilon$-flexible for list size $5$., Comment: 18 pages, 4 figures

Tension pneumothorax is a life-threatening condition that can develop when either the visceral pleura is disrupted, or with injury to the tracheobronchial tree. Rapid, accurate diagnosis and appropriate management are required to prevent significant atelectasis, hypoxia, circulatory arrest, and ultimate patient demise. Needle decompression is the current standard of care for the management of tension pneumothorax. Healthcare providers struggle to assess the success of decompression due to a lack of any immediate objective feedback. The gaseous composition of tension pneumothorax is similar to that of end respiratory gas. This includes an increased partial pressure of carbon dioxide in comparison to atmospheric air, which makes colorimetric capnography an ideal confirmatory test. This colorimetric capnography device may help the healthcare providers to make an objective and accurate assessment of the success of the needle decompression, in particular in prehospital environments.

We examine a hypothesis implied by Steffe's constructivist model of children's numerical reasoning: a child's spontaneous additive strategy may relate to a foundational form of multiplicative reasoning, termed multiplicative double counting (mDC). To this end, we mix quantitative and qualitative analyses of 31 fourth graders' responses during clinical, task-based interviews. All participants spontaneously used one of three additive strategies-counting-on, doubling, or break-apart-make-ten (BAMT)-to correctly solve an addition word problem (8 + 7). We found between-group differences, with asymmetric association of those ordinal variables. We found counting-on to be mainly related to premultiplicative reasoning and BAMT to mDC reasoning. We discuss the theoretical significance and implications of this corroboration of Steffe's model. Colorado Multiple Institutional Review Board (COMIRB); US National Science FoundationNational Science Foundation (NSF) [1503206] This study was approved by the Colorado Multiple Institutional Review Board (COMIRB). We have no conflicts of interest to disclose. Our work was funded in part by the US National Science Foundation under grant #1503206. The opinions expressed do not necessarily reflect the views of the Foundation.

Introduction Plasma thrombin generation kinetics as measured by the calibrated automated thrombogram (CAT) assay is a predictor of symptomatic venous thromboembolism after trauma. We hypothesized that data from a new prototype assay for measurement of thrombin generation kinetics in fresh whole blood (near patient testing of thrombin generation), will correlate with the standard CAT assay in the same patients, making it a potential tool in the future care of trauma patients. Methods Patients were enrolled from June 2018 to February 2020. Within 12 hours of injury, blood samples were collected simultaneously for both assays. Variables compared and correlated between assays were lag time, peak height, time to peak, and endogenous thrombin potential. Data are presented as median with interquartile range (IQR). Spearman and Pearson correlations were estimated and tested between both assays; a P value of

Background Guidelines promote shared decision‐making (SDM) for anticoagulation in patients with atrial fibrillation. We recently showed that adding a within‐encounter SDM tool to usual care (UC) increases patient involvement in decision‐making and clinician satisfaction, without affecting encounter length. We aimed to estimate the extent to which use of an SDM tool changed adherence to the decided care plan and clinical safety end points. Methods and Results We conducted a multicenter, encounter‐level, randomized trial assessing the efficacy of UC with versus without an SDM conversation tool for use during the clinical encounter (Anticoagulation Choice) in patients with nonvalvular atrial fibrillation considering starting or reviewing anticoagulation treatment. We conducted a chart and pharmacy review, blinded to randomization status, at 10 months after enrollment to assess primary adherence (proportion of patients who were prescribed an anticoagulant who filled their first prescription) and secondary adherence (estimated using the proportion of days for which treatment was supplied and filled for direct oral anticoagulant, and as time in therapeutic range for warfarin). We also noted any strokes, transient ischemic attacks, major bleeding, or deaths as safety end points. We enrolled 922 evaluable patient encounters (Anticoagulation Choice=463, and UC=459), of which 814 (88%) had pharmacy and clinical follow‐up. We found no differences between arms in either primary adherence (78% of patients in the SDM arm filled their first prescription versus 81% in UC arm) or secondary adherence to anticoagulation (percentage days covered of the direct oral anticoagulant was 74.1% in SDM versus 71.6% in UC; time in therapeutic range for warfarin was 66.6% in SDM versus 64.4% in UC). Safety outcomes, mostly bleeds, occurred in 13% of participants in the SDM arm and 14% in the UC arm. Conclusions In this large, randomized trial comparing UC with a tool to promote SDM against UC alone, we found no significant differences between arms in primary or secondary adherence to anticoagulation or in clinical safety outcomes. Registration URL: https://www.clinicaltrials.gov ; Unique identifier: clinicaltrials.gov. Identifier: NCT02905032.