Small Ramsey numbers for books, wheels, and generalizations

In this work, we give several new upper and lower bounds on Ramsey numbers for books and wheels, including a tight upper bound establishing $R(W_5, W_7) = 15$, matching upper and lower bounds giving $R(W_5, W_9) = 18$, $R(B_2, B_8) = 21$, and $R(B_3, B_7) = 20$, and a number of additional tight lower bounds for books. We use a range of different methods: flag algebras, local search, bottom-up generation, and enumeration of polycirculant graphs. We also explore generalized Ramsey numbers using similar methods. Let $GR(r,K_s,t)$ denote the minimum number of vertices $n$ such that any $r$-edge-coloring of $K_n$ has a copy of $K_s$ with at most $t$ colors. We establish $GR(3,K_4,2) = 10, GR(4,K_4,3) = 10$, and some additional bounds.

💡 Research Summary

The paper “Small Ramsey numbers for books, wheels, and generalizations” investigates exact and near‑exact Ramsey numbers for two families of graphs—books (B_k) and wheels (W_k)—and extends the study to a generalized Ramsey function (GR(r,K_s,t)). A book (B_k) consists of a spine edge together with (k) “pages” adjacent only to the two spine vertices, while a wheel (W_k) is a cycle of length (k-1) plus a universal hub. The authors combine several computational techniques—flag algebras, tabu search, bottom‑up generation, and exhaustive enumeration of polycirculant graphs—to obtain new upper and lower bounds, many of which are tight.

Key results for classical two‑color Ramsey numbers include:

• (R(W_5,W_7)=15), improving the previous upper bound of 16 and matching the known lower bound.
• (R(W_5,W_9)=18), establishing both upper and lower bounds at 18 (previously 17/17).
• (R(B_2,B_8)=21) and (R(B_3,B_7)=20), both proved exactly.
• Additional tight bounds for many diagonal and near‑diagonal book pairs, summarized in Lemma 1: for (4\le n\le21), (4n-3\le R(B_{n-2},B_n)), (R(B_{n-1},B_n)=4n-1), and (4n+1\le R(B_n,B_n)\le4n+2).

The paper also provides a suite of new bounds for other book pairs (e.g., (R(B_5,B_6)=23), (R(B_6,B_7)=27), up to (R(B_8,B_8)=33)) and improves several previously known estimates.

Methodologically, the authors employ:

1. Flag algebras: Using the semi‑definite programming framework of Razborov, they translate the Ramsey problem into a sum‑of‑squares certificate. The approach is adapted from their earlier work (LP21) by “blowing up” small Ramsey graphs to an asymptotic setting, allowing the derivation of tight upper bounds for most cases (except (R(B_2,B_8)), which required a different treatment).
2. Tabu search: To construct lower‑bound colorings, a simple yet effective tabu‑search algorithm is used. Starting from a random 2‑edge‑coloring of (K_{n-1}), the algorithm iteratively recolors a single edge to minimize a score (the number of forbidden monochromatic substructures). A full‑graph hash table records previously visited colorings, ensuring the search never revisits a state. For books, the score update is computed in (O(|V|)) time by counting common neighborhoods of spine vertices; for wheels the cost is higher but still manageable.
3. Bottom‑up generation: The authors extend the nauty program’s geng tool with a custom plugin that forbids the appearance of wheels or books. By iteratively adding vertices and pruning illegal extensions, they exhaustively generate all Ramsey‑critical graphs for the targeted parameters. This method yielded (R(W_5,W_7)) in 90 seconds and (R(W_5,W_9)) after 36 CPU‑hours; the computation of (R(B_2,B_8)) required roughly two years of CPU time.
4. Polycirculant graph enumeration: Many extremal Ramsey graphs exhibit high symmetry. The authors generate all (k)-polycirculant graphs (graphs admitting an automorphism with all orbits of size (n/k)) that avoid the forbidden subgraphs. For instance, the unique extremal ((B_2,B_8))-graph on 20 vertices is 4‑polycirculant with 240 automorphisms. They also catalog all 2‑polycirculant ((B_2,B_9))-graphs (seven non‑isomorphic) and the single 3‑polycirculant ((B_2,B_{10}))-graph on 24 vertices.

The paper further explores the generalized Ramsey function (GR(r,K_s,t)), defined as the smallest (n) such that any (r)-edge‑coloring of (K_n) contains a copy of (K_s) using at most (t) colors. Using similar computational tools, the authors determine:

• (GR(3,K_4,2)=10) and (GR(4,K_4,3)=10) exactly.
• Upper and lower bounds for other parameters, e.g., (GR(3,K_5,2)=20) (upper bound 23), (GR(3,K_6,2)=32) (upper bound 54), and (GR(4,K_4,2)=15) (upper bound 17).

Explicit constructions are described: for (GR(3,K_4,2)) the extremal coloring consists of a 9‑vertex Paley graph in one color and two disjoint triangles in the other colors; for (GR(3,K_5,2)) a 19‑vertex coloring uses three circulant graphs (C(19,