Talagrand's conjecture that fractional and integral expectation thresholds agree up to a constant

Theorem 1.1: $q_f(\mathcal F)\le25\cdot512^4\,q(\mathcal F)$ for every finite nonempty $X$ and nonempty proper increasing $\mathcal F\subseteq2^X$, independently of $|X|$, the sizes of minimal members and the support of the fractional cover; equivalently every fractional cover of cost at most $1/2$ at $r$ rounds to an integral cover of cost at most $1/2$ at $r/(25\cdot512^4)$. Previously the best general loss was of order $\log\log|X|$ (Fischer-Person with Pham). The constant is explicit but not optimized.

Theorem 1.1: $q_f(\mathcal F)\le25\cdot512^4\,q(\mathcal F)$ for every finite nonempty $X$ and nonempty proper increasing $\mathcal F\subseteq2^X$, independently of $|X|$, the sizes of minimal members and the support of the fractional cover; equivalently every fractional cover of cost at most $1/2$ at $r$ rounds to an integral cover of cost at most $1/2$ at $r/(25\cdot512^4)$. Previously the best general loss was of order $\log\log|X|$ (Fischer-Person with Pham). The constant is explicit but not optimized.