On March 16, 2023, Paul Seymour from Princeton University gave an online talk at the Virtual Discrete Math Colloquium on the existence of a clique or an independent set of size at least
Paul Seymour, A loglog step towards the Erdős-Hajnal conjecture
In 1977, Erdős and Hajnal made the conjecture that, for every graph
There has no improvement on this result (for general
Joint work with Matija Bucić, Tung Nguyen and Alex Scott.
Paul Seymour gave an online talk on the polynomial 𝜒-boundedness of graph classes at the Virtual Discrete Math Colloquium
On October 8, 2021, Paul Seymour from Princeton University gave an online talk at the Virtual Discrete Math Colloquium on the polynomial 𝜒-boundedness of graph classes. The title of his talk was “Polynomial bounds for chromatic number“.
Paul Seymour, Polynomial bounds for chromatic number
The Gyárfás-Sumner conjecture says that for every forest
Nevertheless, there is a much stronger conjecture, due to Esperet: that for every forest
A third notorious conjecture is the Erdős-Hajnal conjecture, that for every graph
Paul Seymour gave an online talk on the recent result regarding the Erdős-Hajnal conjecture at the Virtual Discrete Math Colloquium
On December 30, 2020, Paul Seymour from Princeton University was the speaker of the Virtual Discrete Math Colloquium. He presented his recent breakthrough on the Erdős-Hajnal conjecture with Maria Chudnovsky, Alex Scott, and Sophie Spirkl, which in particular proves that the Erdős-Hajnal conjecture holds for the cycle of length 5. The title of his talk was “The Erdős-Hajnal conjecture is true for excluding a five-cycle“.
Paul Seymour, The Erdős-Hajnal conjecture is true for excluding a five-cycle
In an n-vertex graph, there must be a clique or stable set of size at least
Erdős and Hajnal conjectured in 1977 that for every graph H, there exists c>0 such that every H-free graph has a clique or stable set of size at least
It is true in that case. We will give a sketch of the proof, which is via applying a lemma about bipartite graphs, a variant of a theorem of I. Tomon.
This lemma has several other applications to the Erdős-Hajnal conjecture. For instance, it implies that for every cycle C and forest T, there exists c>0 such that every graph that is both C-free and T’-free (where T’ is the complement of T) has a clique or stable set of size
Joint work with Maria Chudnovsky, Alex Scott and Sophie Spirkl.