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We prove the following 30 year-old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C1,. . .,Cℓ of orders two and three such that |C1|+···+|Cℓ| ≤ (1/2+o(1))n2. This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n2/4.
We estimated the prevalence of food insecurity among people who inject drugs (PWID) in Los Angeles and San Francisco and explored correlates of food insecurity.
A cross-sectional study that measured 30 d food insecurity using the US Adult Food Security Survey ten-item Module developed by the US Department of Agriculture. Food insecurity was defined as including low and very low food security.
Two cities in the state of California, USA.
Male and female active PWID (n 777).
Among participants, 58 % reported food insecurity and 41 % reported very low food security. Food-insecure PWID were more likely to report being homeless (prevalence ratio (PR)=1·20; 95 % CI 1·05, 1·37), chest pain in the past 12 months (PR=1·19; CI 1·06, 1·35), acquiring syringes from someone who goes to a syringe exchange programme (PR=1·27; 95 % CI 1·13, 1·43) and feeling at risk for arrest for possession of drug paraphernalia (PR=1·30; 95 % CI 1·15, 1·46).
Current food insecurity was common among PWID in these two cities, yet few factors were independently associated with food insecurity. These data suggest that broad strategies to improve food access for this high-risk population are urgently needed.
This volume contains nine survey articles based on the invited lectures given at the 25th British Combinatorial Conference, held at the University of Warwick in July 2015. This biennial conference is a well-established international event, with speakers from around the world. The volume provides an up-to-date overview of current research in several areas of combinatorics, including graph theory, Ramsey theory, combinatorial geometry and curves over finite fields. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of cutting-edge developments. The book provides a valuable survey of the present state of knowledge in combinatorics, and will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.
Nešetřil and Ossona de Mendez introduced the notion of first-order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether, if (Gi)i∈ℕ is a sequence of graphs with M being their first-order limit and v is a vertex of M, then there exists a sequence (vi)i∈ℕ of vertices such that the graphs Gi rooted at vi converge to M rooted at v. We show that this holds for almost all vertices v of M, and we give an example showing that the statement need not hold for all vertices.
To identify predictors of good outcome in acute basilar artery occlusion (BAO).
Acute ischemic stroke (AIS) caused by BAO is often associated with a severe and persistent neurological deficit and a high mortality rate.
The set consisted of 70 consecutive AIS patients (51 males; mean age 64.5±14.5 years) with BAO. The role of the following factors was assessed: baseline characteristics, stroke risk factors, pre-event antithrombotic treatment, neurological deficit at time of treatment, estimated time to therapy procedure initiation, treatment method, recanalization rate, change in neurological deficit, post-treatment imaging findings. 30- and 90-day outcome was assessed using the modified Rankin scale with a good outcome defined as a score of 0–3.
The following statistically significant differences were found between patients with good versus poor outcomes: mean age (54.2 vs. 68.9 years; p=0.0001), presence of arterial hypertension (52.4% vs. 83.7%; p=0.015), diabetes mellitus (9.5% vs. 55.1%; p=0.0004) and severe stroke (14.3% vs. 65.3%; p=0.0002), neurological deficit at time of treatment (14.0 vs. 24.0 median of National Institutes of Health Stroke Scale [NIHSS] points; p=0.001), successful recanalization (90.0% vs. 54.2%; p=0.005), change in neurological deficit (12.0 vs. 1.0 median difference of NIHSS points; p=0.005). Stepwise binary logistic regression analysis identified age (OR=0.932, 95% CI=0.882–0.984; p=0.012), presence of diabetes mellitus (OR=0.105, 95% CI=0.018–0.618; p=0.013) and severe stroke (OR=0.071, 95% CI=0.013–0.383; p=0.002) as significant independent negative predictors of good outcome.
In the present study, higher age, presence of diabetes mellitus and severe stroke were identified as significant independent negative predictors of good outcome.
We show that any n-vertex complete graph with edges coloured with three colours contains a set of at most four vertices such that the number of the neighbours of these vertices in one of the colours is at least 2n/3. The previous best value, proved by Erdős, Faudree, Gould, Gyárfás, Rousseau and Schelp in 1989, is 22. It is conjectured that three vertices suffice.
A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Št'ovíček and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colourable.
It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration
. We find three configurations such that a Steiner triple system is affine if and only if it does not contain one of these configurations. Similarly, we characterise Hall triple systems using two forbidden configurations.
Our characterisations have several interesting corollaries in the area of edge-colourings of graphs. A cubic graph
-edge-colourable for a Steiner triple system
if its edges can be coloured with points of
in such a way that the points assigned to three edges sharing a vertex form a triple in
. Among others, we show that all cubic graphs are
-edge-colourable for every non-projective non-affine point-transitive Steiner triple system
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