Multicolor Ramsey numbers and restricted Tur\'an numbers for the loose 3-uniform path of length three
Eliza Jackowska, Joanna Polcyn, Andrzej Ruci\'nski

TL;DR
This paper refines the understanding of multicolor Ramsey numbers for a specific 3-uniform hypergraph by analyzing Turán numbers under restrictions, confirming the formula for certain values of r.
Contribution
It determines the maximum edges in non-star P-free 3-graphs, extending the known Ramsey number results for r=4 to 7.
Findings
Confirmed R(P;r)=r+6 for r=4,5,6,7.
Determined maximum edges in non-star P-free 3-graphs.
Analyzed Turán numbers under additional restrictions.
Abstract
Let denote a 3-uniform hypergraph consisting of 7 vertices and 3 edges and . It is known that the -colored Ramsey number for is for , and that for all . The latter result follows by a standard application of the Tur\'an number , which was determined to be in our previous work. We have also shown that the full star is the only extremal 3-graph for . In this paper, we perform a subtle analysis of the Tur\'an numbers for under some additional restrictions. Most importantly, we determine the largest number of edges in an -vertex -free 3-graph which is not a star. These Tur\'an type results, in turn, allow us to confirm the formula for .
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.
Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
