Proving a conjecture of Talagrand, a fractional version of the``expectation-threshold"
conjecture of Kalai and the second author, we show that p_c(F)=O(q_f(F)log\ell(F)) for any …

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We study the model G α∪ G (n, p) of randomly perturbed dense graphs, where G α is any n‐
vertex graph with minimum degree at least α n and G (n, p) is the binomial random graph …

In the model of randomly perturbed graphs we consider the union of a deterministic graph
$\mathcal {G} _\alpha $ with minimum degree $\alpha n $ and the binomial random graph …

For a family of graphs $\mathcal {F} $, a graph $ G $ is $\mathcal {F} $-universal if $ G $
contains every graph in $\mathcal {F} $ as a (not necessarily induced) subgraph. For the …

The bandwidth theorem [Mathematische Annalen, 343 (1): 175--205, 2009] states that any $
n $-vertex graph $ G $ with minimum degree $\big (\tfrac {k-1}{k}+ o (1)\big) n $ contains all …

In recent years there has been much progress in graph theory on questions of the following
type. What is the threshold for a certain large substructure to appear in a random graph …

A graph is said to be‐universal if it contains every graph with n vertices and maximum
degree at most Δ as a subgraph. Dellamonica, Kohayakawa, Rödl and Ruciński used a …

A graph G is called universal for a family of graphs F if it contains every element F∈ F as a
subgraph. Let F (n, 2) be the family of all graphs with maximum degree 2. Ferber et al.(2019) …

Let r≥ 4 be an integer and consider the following game on the complete graph K n for n∈ r
Z: Two players, Maker and Breaker, alternately claim previously unclaimed edges of K n …