A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. We prove for the unweighted case that this is a.a.s. true when the support is a random hypergraph.

Given a dense subset $A$ of the first $n$ positive integers, we provide a short proof showing that for $p=\omega(n^{-2/3}),$ the so-called randomly perturbed set $A \cup [n]_p$ a.a.s. has the property that any 2-coloring of it has a monochromatic Schur triple, i.e., a triple of the form $(a,b,a+b)$. This result is optimal since there are dense sets $A$, for which $A\cup [n]_p$ does not possess this property for $p=o(n^{-2/3})$.

The hypergraph of the Fano plane is the unique 3-uniform hypergraph with 7 triples on 7 vertices in which every pair of vertices is contained in a unique triple. This hypergraph is not 2-colorable, but becomes so on deleting any hyperedge from it. We show that taking uniformly at random a labeled 3-uniform hypergraph H on n vertices not containing the hypergraph of the Fano plane, H turns out to be 2-colorable with probability at least 1 – 2−Ω(n2). For the proof of this result we will study structural properties of Fano-free hypergraphs.

We study sufficient $\ell$-degree ($1\leq\ell