A sharp version of Talagrand's selector process conjecture and an application to rounding fractional covers
Huy Tuan Pham
TL;DR
The paper tackles the gap between integral and fractional expectation thresholds for monotone properties by proving that, when a fractional cover is supported on sets of size at most $t$, one can round to an integral cover at density $q = c p / \log t$ for an absolute constant $c>0$. This yields a strong bound on the integral threshold from a bounded-size fractional solution, effectively verifying Talagrand's conjecture in the bounded-size regime. The core technical advance is a sharp version of Talagrand's selector process, implemented via towers of minimum fragments, which enables iterative amplification and a robust rounding mechanism from fractional to integral covers. The results also imply a KK-type corollary in the bounded-size setting and open avenues for applying the towers framework to broader threshold problems in random combinatorial structures.
Abstract
Expectation thresholds arise from a class of integer linear programs (LPs) that are fundamental to the study of thresholds in large random systems. An avenue towards estimating expectation thresholds comes from the fractional relaxation of these integer LPs, which yield the fractional expectation thresholds. Regarding the gap between the integer LPs and their fractional relaxations, Talagrand made a bold conjecture, that the integral and fractional expectation thresholds are within a constant factor of each other. In other words, any small fractional solution can be ``rounded''. In this paper, we prove a strong upper bound on the expectation threshold starting from a fractional solution supported on sets with small size. In particular, this resolves Talagrand's conjecture for fractional solutions supported on sets with bounded size. Our key input for rounding the fractional solutions is a sharp version of Talagrand's selector process conjecture that is of independent interest.