The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs $K_{r,r}$ with $r \ge 5$. When H is a complete graph $K_s$ with $s \ge 5$ we show that for some C>0, with high probability the independence number of G is at most $Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}$. This gives new lower bounds for Ramsey numbers R(s,t) for fixed $s \ge 5$ and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.Comment: 36 page

Effects of alirocumab on cardiovascular and metabolic outcomes after acute coronary syndrome in patients with or without diabetes: a prespecified analysis of the ODYSSEY OUTCOMES randomised controlled trial

Background: After acute coronary syndrome, diabetes conveys an excess risk of ischaemic cardiovascular events. A reduction in mean LDL cholesterol to 1·4–1·8 mmol/L with ezetimibe or statins reduces cardiovascular events in patients with an acute coronary syndrome and diabetes. However, the efficacy and safety of further reduction in LDL cholesterol with an inhibitor of proprotein convertase subtilisin/kexin type 9 (PCSK9) after acute coronary syndrome is unknown. We aimed to explore this issue in a prespecified analysis of the ODYSSEY OUTCOMES trial of the PCSK9 inhibitor alirocumab, assessing its effects on cardiovascular outcomes by baseline glycaemic status, while also assessing its effects on glycaemic measures including risk of new-onset diabetes. Methods: ODYSSEY OUTCOMES was a randomised, double-blind, placebo-controlled trial, done at 1315 sites in 57 countries, that compared alirocumab with placebo in patients who had been admitted to hospital with an acute coronary syndrome (myocardial infarction or unstable angina) 1–12 months before randomisation and who had raised concentrations of atherogenic lipoproteins despite use of high-intensity statins. Patients were randomly assigned (1:1) to receive alirocumab or placebo every 2 weeks; randomisation was stratified by country and was done centrally with an interactive voice-response or web-response system. Alirocumab was titrated to target LDL cholesterol concentrations of 0·65–1·30 mmol/L. In this prespecified analysis, we investigated the effect of alirocumab on cardiovascular events by glycaemic status at baseline (diabetes, prediabetes, or normoglycaemia)—defined on the basis of patient history, review of medical records, or baseline HbA1c or fasting serum glucose—and risk of new-onset diabetes among those without diabetes at baseline. The primary endpoint was a composite of death from coronary heart disease, non-fatal myocardial infarction, fatal or non-fatal ischaemic stroke, or unstable angina requiring hospital admission. ODYSSEY OUTCOMES is registered with ClinicalTrials.gov, number NCT01663402. Findings: At study baseline, 5444 patients (28·8%) had diabetes, 8246 (43·6%) had prediabetes, and 5234 (27·7%) had normoglycaemia. There were no significant differences across glycaemic categories in median LDL cholesterol at baseline (2·20–2·28 mmol/L), after 4 months' treatment with alirocumab (0·80 mmol/L), or after 4 months' treatment with placebo (2·25–2·28 mmol/L). In the placebo group, the incidence of the primary endpoint over a median of 2·8 years was greater in patients with diabetes (16·4%) than in those with prediabetes (9·2%) or normoglycaemia (8·5%); hazard ratio (HR) for diabetes versus normoglycaemia 2·09 (95% CI 1·78–2·46, p<0·0001) and for diabetes versus prediabetes 1·90 (1·65–2·17, p<0·0001). Alirocumab resulted in similar relative reductions in the incidence of the primary endpoint in each glycaemic category, but a greater absolute reduction in the incidence of the primary endpoint in patients with diabetes (2·3%, 95% CI 0·4 to 4·2) than in those with prediabetes (1·2%, 0·0 to 2·4) or normoglycaemia (1·2%, −0·3 to 2·7; absolute risk reduction pinteraction=0·0019). Among patients without diabetes at baseline, 676 (10·1%) developed diabetes in the placebo group, compared with 648 (9·6%) in the alirocumab group; alirocumab did not increase the risk of new-onset diabetes (HR 1·00, 95% CI 0·89–1·11). HRs were 0·97 (95% CI 0·87–1·09) for patients with prediabetes and 1·30 (95% CI 0·93–1·81) for those with normoglycaemia (pinteraction=0·11). Interpretation: After a recent acute coronary syndrome, alirocumab treatment targeting an LDL cholesterol concentration of 0·65–1·30 mmol/L produced about twice the absolute reduction in cardiovascular events among patients with diabetes as in those without diabetes. Alirocumab treatment did not increase the risk of new-onset diabetes. Funding: Sanofi and Regeneron Pharmaceuticals

Lipoprotein(a) and Benefit of PCSK9 Inhibition in Patients With Nominally Controlled LDL Cholesterol

Background: Guidelines recommend nonstatin lipid-lowering agents in patients at very high risk for major adverse cardiovascular events (MACE) if low-density lipoprotein cholesterol (LDL-C) remains ≥70 mg/dL on maximum tolerated statin treatment. It is uncertain if this approach benefits patients with LDL-C near 70 mg/dL. Lipoprotein(a) levels may influence residual risk. Objectives: In a post hoc analysis of the ODYSSEY Outcomes (Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab) trial, the authors evaluated the benefit of adding the proprotein subtilisin/kexin type 9 inhibitor alirocumab to optimized statin treatment in patients with LDL-C levels near 70 mg/dL. Effects were evaluated according to concurrent lipoprotein(a) levels. Methods: ODYSSEY Outcomes compared alirocumab with placebo in 18,924 patients with recent acute coronary syndromes receiving optimized statin treatment. In 4,351 patients (23.0%), screening or randomization LDL-C was 13.7 mg/dL or ≤13.7 mg/dL; corresponding adjusted treatment hazard ratios were 0.82 (95% CI: 0.72-0.92) and 0.89 (95% CI: 0.75-1.06), with Pinteraction = 0.43. Conclusions: In patients with recent acute coronary syndromes and LDL-C near 70 mg/dL on optimized statin therapy, proprotein subtilisin/kexin type 9 inhibition provides incremental clinical benefit only when lipoprotein(a) concentration is at least mildly elevated. (ODYSSEY Outcomes: Evaluation of Cardiovascular Outcomes After an Acute Coronary Syndrome During Treatment With Alirocumab; NCT01663402