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.

翻译：期望阈值源于一类整数线性规划（LP），这类规划对于研究大型随机系统中的阈值至关重要。估计期望阈值的一种途径来自这些整数线性规划的分数松弛，从而得到分数期望阈值。关于整数线性规划与其分数松弛之间的差距，Talagrand提出了一个大胆的猜想：整数期望阈值与分数期望阈值彼此相差一个常数因子。换言之，任何小的分数解都可以被“舍入”。本文中，我们从支撑集规模较小的分数解出发，证明了期望阈值的一个强上界。特别地，这解决了支撑集规模有界的分数解对应的Talagrand猜想。我们舍入分数解的关键输入是Talagrand选择过程猜想的一个精确形式，该形式本身具有独立的研究价值。