You are cordially invited to our next Monday Lecture &
Colloquium on June 21st. Time is back to normal. Monday's Lecture & Colloquium will be online via Zoom, as used to. Invitation link: Zoom - Invitation Time: Monday, June 21st - 14:15 h Lecture: Alexey Pokrovskiy (University College London) Title: Linear size Ramsey numbers of hypergraphs Abstract:The size-Ramsey number of a hypergraph H is the minimum number of edges in a hypergraph G whose every 2-edge-colouring contains a monochromatic copy of H. This talk will be about showing that the size-Ramsey number of r-uniform tight path on n vertices is linear in n. Similar results about hypergraph trees and their powers will also be discussed. This is joint work with Letzter and Yepremyan. Coffee Break! Time: Monday, June 21st - 16:00 h s.t. Colloquium: Patrick Morris (Freie Universität Berlin) Title: Triangle factors in pseudorandom graphs Abstract:An (n, d, λ)-graph is an n vertex, d-regular graph with second eigenvalue in absolute value λ. When λ is small compared to d, such graphs have pseudo-random properties. A fundamental question in the study of pseudorandom graphs is to find conditions on the parameters that guarantee the existence of a certain subgraph. A celebrated construction due to Alon gives a triangle-free (n, d, λ)-graph with d = Θ(n^2/3) and λ = Θ(d^2/n). This construction is optimal as having λ = o(d^2/n) guarantees the existence of a triangle in a (n, d, λ)-graph. Krivelevich, Sudakov and Szab ́o (2004) conjectured that if n ∈ 3N and λ = o(d^2/n) then an (n, d, λ)-graph G in fact contains a triangle factor: vertex disjoint triangles covering the whole vertex set. In this talk, we discuss a solution to the conjecture of Krivelevich, Sudakov and Szab ́o. The result can be seen as a clear distinction between pseudorandom graphs and random graphs, showing that essentially the same pseudorandom condition that ensures a triangle in a graph actually guarantees a triangle factor. In fact, even more is true: as a corollary to this result and a result of Han, Kohayakawa, Person and the author, we can conclude that the same condition actually guarantees that such a graph G contains every graph on n vertices with maximum degree at most 2. |