An Improved Lower Bound for $S(7)$ and Some Interesting Templates
Fred Rowley

TL;DR
This paper presents a new lower bound for the Schur number $S(7)$ using a specific partition, and introduces a construction that improves bounds for larger Schur numbers and multicolour Ramsey numbers.
Contribution
The paper provides a new explicit partition that establishes $S(7) ext{ge} 1696$ and introduces a recursive construction to bound larger Schur numbers and Ramsey numbers.
Findings
Established $S(7) ext{ge} 1696$ using a Schur partition.
Derived a recursive inequality $S(k+5) ext{ge} 376.S(k)+160$.
Showed that $ ext{lim}_{r o \infty} R_r(3)^{1/r} > 3.273$.
Abstract
In this simple paper, we exhibit a Schur partition giving rise to a triangle-free linear colouring of in 7 colours. Thus we show that the Schur number and the multicolour Ramsey number . We also demonstrate a specific partition of the closed integer interval which, through repeated use of a specific construction, allows us to conclude that for any positive integer . The existence of this partition, with its specific properties, implies that .
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
Topicssemigroups and automata theory · Limits and Structures in Graph Theory · Computability, Logic, AI Algorithms
