The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $\epsilon \gt 0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^{\epsilon }$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

Men who have sex with men (MSM) who use injection drugs (MSM-IDU) are at high risk of sexually transmitted infections (STIs), but the long-term incidence is unclear. We conducted a single-centre retrospective cohort study using the clinical records of non-haemophilia men with human immunodeficiency virus (HIV) who visited the Institute of Medical Science, the University of Tokyo (IMSUT) Hospital, located in Tokyo, Japan, from 2013 to 2022. We analysed 575 patients including 62 heterosexual males and 513 MSM patients, of whom 6.8% (35/513) were injection drug use (IDU). Compared to non-IDU MSM, MSM-IDU had a higher incidence of hepatitis C virus (HCV) (44.8 vs 3.5 /1,000 person-years (PY); incidence rate ratio (IRR) [95% confidence interval (95% CI)], 12.8 [5.5–29.3], p < 0.001) and syphilis (113.8 vs 53.3 /1,000 PY; IRR, 2.1 [1.4–3.1], p < 0.001). The incidence of other symptomatic STIs (amoebiasis, chlamydia, and gonorrhoea infections) was <4/1,000 PY. In multivariable Poisson regression analysis, HCV incidence was associated with MSM (IRR, 1.8 × 106 [9.9 × 105–3.4 × 106], p < 0.001), IDU (IRR, 10.1 [4.0–25.6], p < 0.001), and syphilis infection during the study period (IRR, 25.0 [1.2–518.3]/time/year, p < 0.001). Among men with HIV, the prevalence of IDU in MSM and the long-term incidence of STIs in MSM-IDU were high. IDU and sexual contact are important modes of transmission of HCV among HIV-infected MSM in Tokyo.

Congenital Zika is a devastating consequence of maternal Zika virus infections. Estimates of age-dependent seroprevalence profiles are central to our understanding of the force of Zika virus infections. We set out to calculate the age-dependent seroprevalence of Zika virus infections in Brazil. We analyzed serum samples stratified by age and geographic location, collected from 2016 to 2019, from about 16,000 volunteers enrolled in a Phase 3 dengue vaccine trial led by the Institute Butantan in Brazil. Our results show that Zika seroprevalence has a remarkable age-dependent and geographical distribution, with an average age of the first infection varying from region to region, ranging from 4.97 (3.03–5.41) to 7.24 (6.98–7.90) years. The calculated basic reproduction number, $ {R}_0 $, varied from region to region, ranging from 1.18 (1.04–1.41) to 2.33 (1.54–3.85). Such data are paramount to determine the optimal age to vaccinate against Zika, if and when such a vaccine becomes available.