[Paper Review] Multicolor Ramsey numbers and restricted Tur\'an numbers for the loose 3-uniform path of length three

Eliza Jackowska, Joanna Polcyn+1 more|arXiv (Cornell University)|Jul 15, 2015

Limits and Structures in Graph Theory9 references3 citations

TL;DR

This paper determines the multicolor Ramsey number for the loose 3-uniform path of length three, proving R(P; r) = r + 6 for all r ≤ 7 by refining Turán-type results. It introduces higher-order Turán numbers—specifically, the maximum number of edges in a P-free 3-graph that is not a star—enabling tighter bounds on Ramsey numbers through structural analysis of extremal hypergraphs.

ABSTRACT

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Tur\'an number $ex_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Tur\'an numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Tur\'an type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.

Motivation & Objective

• To determine the exact multicolor Ramsey number R(P; r) for the loose 3-uniform path P of length three.
• To refine classical Turán number bounds by introducing the concept of second- and third-order Turán numbers for P.
• To establish that the only extremal P-free 3-graphs not contained in a star are the comet hypergraphs Co(n), which maximize edge count under non-star constraints.
• To resolve the Ramsey number for r = 4 to 7, confirming R(P; r) = r + 6 in these cases.

Proposed method

• Introduces the notion of Turán numbers of the s-th order, specifically for P, to capture the maximum edge count in P-free 3-graphs that are not subsets of any star.
• Uses induction on n ≥ 11, with base cases n = 8, 9, 10, to bound edge counts in P- and C-free 3-graphs under structural constraints.
• Applies conditional Turán numbers ex(n; P|M) where M is a fixed substructure (e.g., P₃² ∪ K₃³), and proves extremal configurations are unique comets.
• Employs case analysis based on the size of the set W₀ (vertices not in any edge of the core), leveraging inequalities from earlier lemmas to bound edge counts.
• Leverages known extremal results: ex₃(n; P) = (n−1 choose 2), with the full star S₃ⁿ as the unique extremal hypergraph.
• Uses the fact that if H − v is a comet Co(n−1), and deg(v) = n−5, then H must be a comet Co(n), establishing tightness of the bound.

Experimental results

Research questions

• RQ1 What is the maximum number of edges in an n-vertex P-free 3-graph that is not contained in any star?
• RQ2 Can the Ramsey number R(P; r) be exactly determined for r > 3 using refined Turán-type results?
• RQ3 Is the comet hypergraph Co(n) the unique extremal configuration among non-star P-free 3-graphs?
• RQ4 Does the formula R(P; r) = r + 6 hold for r = 4, 5, 6, 7, and if so, under what structural conditions on edge colorings?

Key findings

• The second-order Turán number ex²(n; P) is shown to be 4 + (n−5 choose 2), achieved uniquely by the comet hypergraph Co(n).
• For r = 4 to 7, the Ramsey number R(P; r) is exactly r + 6, confirmed via structural analysis of edge colorings and Turán-type extremal bounds.
• The only P-free 3-graph on n vertices with more than (n−1 choose 2) edges is the full star S₃ⁿ, and all other extremal P-free 3-graphs not contained in a star are isomorphic to Co(n).
• The bound ex₃(n; P) = (n−1 choose 2) is sharp, and equality holds only for the full star, which is crucial for Ramsey number derivations.
• For n ≥ 11, any P-free 3-graph with more than 4 + (n−5 choose 2) edges must be a comet Co(n), and this structure is necessary to achieve the Ramsey number bound.
• The inductive proof shows that if H − v = Co(n−1) and deg(v) = n−5, then H = Co(n), establishing the tightness of the upper bound on edge count.