Matthew Kwan

Bounded-Degree Spanning Trees in Randomly Perturbed Graphs

We show that for any fixed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into fixed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Specifically, we show that there is

Intercalates and discrepancy in random Latin squares

c = c(\alpha,\Delta)

Non-trivially intersecting multi-part families

such that if G is an n-vertex graph with minimum degree at least

Resilience for the Littlewood-Offord Problem

\alpha n

Almost all Steiner triple systems have perfect matchings

, and T is an n-vertex tree with maximum degree at most

Cycles and matchings in randomly perturbed digraphs and hypergraphs

\Delta

The average number of spanning trees in sparse graphs with given degrees

, then if we add cn uniformly random edges to G, the resulting graph will contain T asymptotically almost surely (as

Ramsey Graphs Induce Subgraphs of Quadratically Many Sizes

n\to\infty

Anticoncentration for subgraph statistics.

). Our proof uses a lemma concerning the decomposition of a dense graph into super-regular pairs of comparable sizes, which may be of independent interest.

Nearly-linear monotone paths in edge-ordered graphs.

An intercalate in a Latin square is a