Thresholds versus fractional expectationthresholds
Abstract
Proving a conjecture of Talagrand, a fractional version of the 'expectationthreshold' conjecture of Kalai and the second author, we show for any increasing family $F$ on a finite set $X$ that $p_c (F) =O( q_f (F) \log \ell(F))$, where $p_c(F)$ and $q_f(F)$ are the threshold and 'fractional expectationthreshold' of $F$, and $\ell(F)$ is the largest size of a minimal member of $F$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (JohanssonKahnVu), boundeddegree spanning trees (Montgomery), and boundeddegree spanning graphs (new). We also resolve (and vastly extend) the 'axial' version of the random multidimensional assignment problem (earlier considered by MartinMézardRivoire and FriezeSorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the ErdősRado 'Sunflower Conjecture'.
 Publication:

arXiv eprints
 Pub Date:
 October 2019
 arXiv:
 arXiv:1910.13433
 Bibcode:
 2019arXiv191013433F
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics;
 Mathematics  Probability
 EPrint:
 16 pages, submitted, now includes some discussion of applications