Sharp threshold

For a sequence of nontrivial increasing events on finite product measures, write $p_x$ for the parameter at which the success probability is $x$. An additive sharp threshold means that $p_{1-\epsilon }-p_{\epsilon }\to 0$ for every fixed $0<\epsilon <1/2$. When the threshold parameter itself tends to zero, relative sharpness instead compares this window with $p_{1/2}$; additive sharpness alone is then a weaker assertion. The Friedgut-Kalai sharp threshold theorem provides an explicit additive bound for transitive increasing events.