Use of MAX-CUT for Ramsey Arrowing of Triangles

In 1967, Erd\H{o}s and Hajnal asked the question: Does there exist a $K_4$-free graph that is not the union of two triangle-free graphs? Finding such a graph involves solving a special case of the classical Ramsey arrowing operation. Folkman proved the existence of these graphs in 1970, and they are now called Folkman graphs. Erd\H{o}s offered $100 for deciding if one exists with less than $10^{10}$ vertices. This problem remained open until 1988 when Spencer, in a seminal paper using probabilistic techniques, proved the existence of a Folkman graph of order $3\times 10^9$ (after an erratum), without explicitly constructing it. In 2008, Dudek and R"{o}dl developed a strategy to construct new Folkman graphs by approximating the maximum cut of a related graph, and used it to improve the upper bound to 941. We improve this bound first to 860 using their approximation technique and then further to 786 with the MAX-CUT semidefinite programming relaxation as used in the Goemans-Williamson algorithm.

💡 Research Summary

The paper addresses a long‑standing problem in Ramsey theory originally posed by Erdős and Hajnal in 1967: determine the smallest order of a K₄‑free graph that cannot be expressed as the union of two triangle‑free graphs. In the language of Ramsey arrowing, this is the Folkman number Fₑ(3,3;4), the minimum number of vertices of a K₄‑free graph G such that every 2‑edge‑coloring of G forces a monochromatic triangle in at least one color.

Historically, Folkman proved existence in 1970, Spencer gave a non‑constructive probabilistic proof in 1988 (later corrected to a graph of order 3 × 10⁹), and Dudek and Rödl (2008) introduced a constructive method based on the maximum‑cut problem. They associated to any candidate graph G a derived graph H_G whose vertices correspond to the edges of G and whose edges connect pairs of edges that belong to a common triangle in G. Theorem 1 (Dudek‑Rödl) states that G → (3,3) if and only if MC(H_G) < 2·t_Δ(G), where t_Δ(G) is the number of triangles in G. This reduction allows one to use tools from combinatorial optimization to test the arrowing property.

Dudek and Rödl applied a simple spectral bound: for any graph H, MC(H) ≤ |E(H)|/2 – λ_min·|V(H)|/4, where λ_min is the smallest eigenvalue of the adjacency matrix of H. Using the circulant graph G(941,5) (941 vertices, 707 632 triangles) they showed λ_min ≈ –14.66, which yields MC(H_G) < 2·t_Δ(G) and thus Fₑ(3,3;4) ≤ 941.

The present authors first improve this bound by removing a carefully chosen set of 81 vertices from G(941,5), obtaining a subgraph G_C with 860 vertices, 73 981 edges, and 5 425 014 triangles. Computing λ_min ≈ –14.663 for H_G_C and applying the same eigenvalue inequality gives MC(H_G_C) < 2·t_Δ(G_C), establishing G_C → (3,3) and consequently Fₑ(3,3;4) ≤ 860.

To push the bound further, the authors employ the Goemans‑Williamson semidefinite programming (SDP) relaxation for MAX‑CUT. The quadratic integer program Max ½∑{i<j} w{ij}(1–y_i y_j) with y_i ∈ {–1,1} is relaxed to Max ½∑{i<j} w{ij}(1–v_i·v_j) subject to ‖v_i‖=1, yielding an SDP whose optimal value is an upper bound on the true MAX‑CUT. Using two large‑scale SDP solvers (SDPLR‑MC and SBmethod) the authors evaluate many candidate graphs, including the family L(n,s) of circulant graphs studied by Lu (2008). For most of these graphs the SDP bound coincides with the eigenvalue bound, confirming the strength of the spectral method.

The breakthrough comes from a modified circulant graph L(785,53). By adding a single new vertex adjacent to a specific set of 60 existing vertices, they construct a graph G_786 with 786 vertices, 6 129 edges, and 42 881 triangles. SDP solvers produce an upper bound MC(H_G_786) ≤ 85 775.3, while a specialized algorithm (SpeeDP) yields a lower bound 85 774.2 ≤ MC(H_G_786) ≤ 85 775.0. Since 2·t_Δ(G_786) = 85 762, both bounds together imply MC(H_G_786) < 2·t_Δ(G_786), i.e., G_786 → (3,3). Consequently, the authors establish the new record Fₑ(3,3;4) ≤ 786.

The paper also discusses open questions. The status of G(127,3) (equivalently L(127,5)) remains unresolved; it is conjectured to arrow (3,3), but neither the eigenvalue nor the SDP approaches have settled it. Moreover, Erdős’s $100 prize for proving Fₑ(3,3;4) < 10¹⁰ and Graham’s $100 prize for a bound below 100 are still open, as the current best upper bound is 786. The lower bound remains at 19, with only a modest improvement to 20 being considered a significant step.

In summary, the authors demonstrate that translating a Ramsey arrowing problem into a MAX‑CUT framework, and then applying both spectral techniques and modern SDP relaxations, yields substantial improvements on the Folkman number. Their work bridges combinatorial extremal graph theory with algorithmic optimization, providing a constructive pathway toward tighter bounds and highlighting promising directions for future research.