Directed graphs with lower orientation Ramsey thresholds
Gabriel Ferreira Barros, Bruno Pasqualotto Cavalar, Yoshiharu, Kohayakawa, Guilherme Oliveira Mota, T\'assio Naia

TL;DR
This paper studies the threshold probability for the Ramsey property in random directed graphs, revealing that the known upper bounds are not always tight and providing specific examples where the threshold differs.
Contribution
It introduces new examples of directed graphs where the classical upper bound for the Ramsey threshold is not tight, especially involving rooted products and certain sparse graphs.
Findings
The upper bound $p_{\vec H}\leq Cn^{-1/m_2(\vec H)}$ is not always the threshold.
Examples include rooted products of orientations of sparse graphs.
Thresholds vary depending on graph structures such as forests and cycles.
Abstract
We investigate the threshold for the Ramsey-type property , where is the binomial random graph and indicates that every orientation of the graph contains the oriented graph as a subdigraph. Similarly to the classical Ramsey setting, the upper bound is known to hold for some constant , where denotes the maximum -density of the underlying graph of . While this upper bound is indeed the threshold for some , this is not always the case. We obtain examples arising from rooted products of orientations of sparse graphs (such as forests, cycles and, more generally, subcubic -free graphs) and arbitrarily rooted transitive triangles.
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
TopicsAdvanced Topology and Set Theory · Limits and Structures in Graph Theory · Advanced Graph Theory Research
