A reformulation of the discrete Convexity Conjecture via $k$-thresholds

We introduce the notion of "$k$-thresholds'' and show that Talagrand's discrete convexity conjecture is equivalent to the assertion that, for some universal integer $k \ge 2$, the $k$-threshold of every increasing family is at most a universal constant times its expectation threshold. We prove a reduction theorem that bounds the $k$-threshold of any increasing graph property in terms of ordinary thresholds of graphs in suitable decompositions of its members. As a consequence, we determine, up to a constant factor, the $k$-threshold of every fixed graph in terms of a natural $k$-density parameter. We also prove that $k=2$ suffices for several classical spanning graph containment properties. More generally, we establish the conjectured comparison between $k$-thresholds and expectation thresholds for broad classes of graph containment properties whose target graphs have low degeneracy.