Ramsey numbers upon vertex deletion
Yuval Wigderson

TL;DR
This paper disproves a conjecture that deleting a vertex from a graph only slightly changes its Ramsey number, by providing a family of graphs where the change is significantly larger, revealing new complexity in Ramsey theory.
Contribution
The authors construct an infinite family of graphs where vertex deletion causes a super-constant decrease in Ramsey numbers, challenging previous conjectures.
Findings
Deleting a vertex can drastically reduce the Ramsey number for certain graphs.
Existence of graphs with colorings where one color class has negligible density.
Disproof of the conjecture by Conlon, Fox, and Sudakov.
Abstract
Given a graph , its Ramsey number is the minimum so that every two-coloring of contains a monochromatic copy of . It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from , the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs so that in any Ramsey coloring for (that is, a coloring of a clique on vertices with no monochromatic copy of ), one of the color classes has density .
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
