We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any $n$-vertex graph $G$ satisfying a given minimum degree condition and the binomial random graph $G(n,p)$. We prove that asymptotically almost surely $G \cup G(n,p)$ contains at least $\min \{\delta (G), \lfloor n/3 \rfloor \}$ pairwise vertex-disjoint triangles, provided $p \ge C \log n/n$, where $C$ is a large enough constant. This is a perturbed version of an old result of Dirac.

Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.

We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and $2$-universality.

Hepatitis B virus-related acute-on-chronic liver failure (HBV-ACLF) is a severe and life-threatening complication, characterised by multi-organ failure and high short-term mortality. However, there is limited information on the impact of various comorbidities on HBV-ACLF in a large population. This study aimed to investigate the relationship between comorbidities, complications and mortality. In this retrospective observational study, we identified 2166 cases of HBV-ACLF hospitalised from January 2010 to March 2018. Demographic data from the patients, medical history, treatment, laboratory indices, comorbidities and complications were collected. The mortality rate in our study group was 47.37%. Type 2 diabetes mellitus was the most common comorbidity, followed by alcoholic liver disease. Spontaneous bacterial peritonitis, pneumonia and hepatic encephalopathy (HE) were common in these patients. Diabetes mellitus and hyperthyroidism are risk factors for death within 90 days, together with gastrointestinal bleeding and HE at admission, HE and hepatorenal syndrome during hospitalisation. Knowledge of risk factors can help identify HBV-ACLF patients with a poor prognosis for HBV-ACLF with comorbidities and complications.

Trustworthiness is typically regarded as a desirable feature of national identification systems (NISs); but the variegated nature of the trustor communities associated with such systems makes it difficult to see how a single system could be equally trustworthy to all actual and potential trustors. This worry is accentuated by common theoretical accounts of trustworthiness. According to such accounts, trustworthiness is relativized to particular individuals and particular areas of activity, such that one can be trustworthy with regard to some individuals in respect of certain matters, but not trustworthy with regard to all trustors in respect of every matter. The present article challenges this relativistic approach to trustworthiness by outlining a new account of trustworthiness, dubbed the expectation-oriented account. This account allows for the possibility of an absolutist (or one-place) approach to trustworthiness. Such an account, we suggest, is the approach that best supports the effort to develop NISs. To be trustworthy, we suggest, is to minimize the error associated with trustor expectations in situations of social dependency (commonly referred to as trust situations), and to be trustworthy in an absolute sense is to assign equal value to all expectation-related errors in all trust situations. In addition to outlining the features of the expectation-oriented account, we describe some of the implications of this account for the design, development, and management of trustworthy NISs.