Doubly Saturated Ramsey-Good Graphs

• The paper introduces doubly saturated R(s,t)-good graphs where adding or deleting any edge destroys the desirable clique and independent set properties.
• A constructive circulant family for R(4,t)-good graphs is provided, affirming classic conjectures and revealing minimal vertex count formulas.
• Computational search, SAT encoding, and LLM-assisted formal verification are key methodologies underpinning the discovery and proof of these graphs.

Doubly saturated Ramsey-good graphs are graphs $G$ that are $R(s,t)$-good in the sense that they contain neither a clique of size $s$ nor an independent set of size $t$, and for which both adding any missing edge and deleting any present edge destroys that property. In the 2026 formulation, one also excludes the complete and empty graphs, so that both edge addition and edge deletion are genuinely possible (Przybocki et al., 23 Apr 2026). The notion strengthens ordinary $R(s,t)$-goodness by requiring simultaneous edge-maximality and edge-minimality inside the class of graphs with $\omega(G)<s$ and $\alpha(G)<t$, and it sits at the intersection of Ramsey theory, graph saturation, computational search, and formal verification (Przybocki et al., 23 Apr 2026).

1. Definition and equivalent formulations

A graph is called $R(s,t)$-good if it contains neither a clique of size $s$ nor an independent set of size $t$. Equivalently,

$R(s,t)$0

A graph $R(s,t)$1 is doubly saturated $R(s,t)$2-good if $R(s,t)$3 is $R(s,t)$4-good, but adding or removing any edge from $R(s,t)$5 yields a graph that is not $R(s,t)$6-good, and neither $R(s,t)$7 nor $R(s,t)$8 is the complete graph (Przybocki et al., 23 Apr 2026).

This definition admits two useful equivalent viewpoints. First, doubly saturated graphs are exactly the $R(s,t)$9-good graphs that are simultaneously edge-maximal and edge-minimal, excluding complete and empty graphs. Second, if one partially orders graphs by edge addition, $s$0 when $s$1 is obtained from $s$2 by adding edges, then doubly saturated graphs are exactly those whose weakly connected component in this poset consists of a single graph (Przybocki et al., 23 Apr 2026).

The local witness formulation is especially important. In the SAT encoding used in the literature, maximality is expressed by saying that every non-edge $s$3 is part of a $s$4, while minimality is expressed by saying that every edge $s$5 is part of an $s$6 (Przybocki et al., 23 Apr 2026). Thus adding a missing edge must complete a forbidden clique, and deleting a present edge must complete a forbidden independent set.

2. Historical emergence and existence theory

The modern study of doubly saturated $s$7-good graphs answers a question with roots in the early 1980s. Albertson and Berman asked in 1980 whether there are any doubly saturated $s$8-good graphs other than $s$9, and Grinstead and Roberts in 1982 found three more examples, all circulant, then asked whether there were any others (Przybocki et al., 23 Apr 2026). Grinstead and Roberts used the term “bicritical,” but the recent literature avoids that term because it is overloaded elsewhere (Przybocki et al., 23 Apr 2026).

The first infinite family is now known. The principal theorem states: $t$0 This is an existential result, but also fully constructive, because the family is given explicitly as circulant graphs (Przybocki et al., 23 Apr 2026). The result answers the Grinstead–Roberts question affirmatively and turns a previously sporadic phenomenon into an infinite theory.

The same work formulates a broader conjecture: $t$1 It also states a lower bound: every doubly saturated $t$2-good graph has at least $t$3 vertices (Przybocki et al., 23 Apr 2026). A further open problem asks whether the $t$4 family is minimal: $t$5 SAT experiments verify this for $t$6 (Przybocki et al., 23 Apr 2026).

3. The circulant $t$7 family

The explicit infinite family is circulant. Let

$t$8

One description is the circulant graph on $t$9 vertices with distance set

$R(s,t)$0

An isomorphic and simpler description, used in the proof, is the circulant graph on $R(s,t)$1 vertices with distances

$R(s,t)$2

The symmetric adjacency criterion is rewritten as

$R(s,t)$3

in $R(s,t)$4 (Przybocki et al., 23 Apr 2026).

The proof of double saturation is organized into four exact conditions. First, the graph has no $R(s,t)$5. This is shown by a cyclic gap argument: if a 4-clique existed, the consecutive gaps around the cycle would have to be either $R(s,t)$6 or at least $R(s,t)$7, and the gap sum $R(s,t)$8 forces a contradiction because two consecutive $R(s,t)$9-gaps would create a non-edge at distance $\omega(G)<s$0 (Przybocki et al., 23 Apr 2026).

Second, the graph has no independent set of size $\omega(G)<s$1. The proof places one vertex at $\omega(G)<s$2, observes that all other vertices of an independent set must lie in

$\omega(G)<s$3

and then uses a pigeonhole-style difference argument to force two vertices at circular distance $\omega(G)<s$4, which is an edge-distance (Przybocki et al., 23 Apr 2026).

Third, maximality is verified by showing that adding any non-edge creates a $\omega(G)<s$5. By symmetry it suffices to add $\omega(G)<s$6, where

$\omega(G)<s$7

If $\omega(G)<s$8, then

$\omega(G)<s$9

forms a $\alpha(G)<t$0 after adding $\alpha(G)<t$1; if $\alpha(G)<t$2, then

$\alpha(G)<t$3

does so (Przybocki et al., 23 Apr 2026).

Fourth, minimality is verified by showing that deleting any edge creates an independent set of size $\alpha(G)<t$4. Again by symmetry one deletes $\alpha(G)<t$5, where

$\alpha(G)<t$6

The proof gives three explicit constructions of independent $\alpha(G)<t$7-sets after deletion, according to whether $\alpha(G)<t$8, $\alpha(G)<t$9, or $R(s,t)$0 (Przybocki et al., 23 Apr 2026).

4. Computational evidence and small examples

The existence theorem emerged from exhaustive and semi-exhaustive computation. The $R(s,t)$1 and $R(s,t)$2 cases were decisive in recognizing the infinite pattern, and the paper records a broader body of evidence for other parameter pairs (Przybocki et al., 23 Apr 2026).

Parameters	Finding	Notes
$R(s,t)$3	Exactly six doubly saturated graphs	Sizes $R(s,t)$4
$R(s,t)$5	None on 18 or fewer vertices; one on 19 vertices	Circulant with distances $R(s,t)$6
$R(s,t)$7	Unique doubly saturated graph	$R(s,t)$8
$R(s,t)$9	No doubly saturated graph	Exception in the conjecture
$s$0	Unique doubly saturated graph	Circulant on 13 vertices with distances $s$1
$s$2	No doubly saturated graph	Exception in the conjecture

Additional findings reinforce the conjectural landscape. The paper reports a doubly saturated $s$3-good graph on 20 vertices, none smaller, likely unique up to isomorphism among found solutions; it is 5-regular, vertex-transitive, not circulant, and decomposable into four disjoint 5-cycles with a matching between each pair of cycles. It also reports two non-isomorphic doubly saturated $s$4-good graphs on 25 vertices, that the unique $s$5-good graph on 35 vertices is automatically doubly saturated and circulant, and that over 100 doubly saturated $s$6-good graphs on 39 vertices were found (Przybocki et al., 23 Apr 2026).

For diagonal parameters, the paper says that Paley graphs show existence of doubly saturated $s$7-good graphs for all $s$8 (Przybocki et al., 23 Apr 2026). This gives substantial computational evidence that doubly saturated behavior is not confined to isolated off-diagonal families.

5. Computer-assisted discovery and formal verification

The 2026 work is also a case study in computer-assisted mathematical discovery. Its SAT encoding uses Boolean edge variables $s$9 for unordered pairs $t$0, together with the standard clauses forbidding $t$1-cliques and $t$2-independent sets (Przybocki et al., 23 Apr 2026). The direct encoding uses $t$3 clauses, which becomes prohibitive as $t$4 grows.

The technical innovation is a compact $t$5-clause encoding of double saturation. For maximality, auxiliary variables $t$6 are introduced to indicate that $t$7 is selected as part of a witness $t$8 for the non-edge $t$9. The constraints require that if $R(s,t)$00 is missing, then at least $R(s,t)$01 selected vertices are adjacent to both $R(s,t)$02 and $R(s,t)$03 and pairwise adjacent to one another. Minimality is encoded analogously for independent sets. Cardinality constraints are implemented using Sinz’s sequential counter encoding, via PySAT (Przybocki et al., 23 Apr 2026).

The search is further accelerated by lexicographic symmetry-breaking constraints on vertex neighborhoods. These reduce isomorphic duplicates and enable exact nonexistence results such as: no doubly saturated $R(s,t)$04-good graph on $R(s,t)$05 or fewer vertices, the smallest doubly saturated $R(s,t)$06-good graph has $R(s,t)$07 vertices, and the smallest doubly saturated $R(s,t)$08-good graph has $R(s,t)$09 vertices (Przybocki et al., 23 Apr 2026).

When the SAT formulations became too large because of the explosion in clauses forbidding $R(s,t)$10-independent sets, the search moved partly to bespoke code generated by an LLM, restricted to circulant graphs. The discovery pipeline was:

1. formulate the problem;
2. generate small-$R(s,t)$11 data using SAT or code;
3. inspect patterns and conjecture a family, with LLM assistance;
4. obtain a proof draft from an LLM;
5. formalize in Lean (Przybocki et al., 23 Apr 2026).

The paper is explicit that ChatGPT played a major role in the conjecture-generation stage for the $R(s,t)$12 family, and that Gemini 3 Deep Think produced a detailed informal proof later fed to Harmonic’s autoformalization system Aristotle, which produced over 1000 lines of Lean code; the formalization succeeded after two attempts (Przybocki et al., 23 Apr 2026). The methodological point is not merely historical. The authors distinguish between experimentally discovered patterns, informal LLM-generated proofs, and fully verified theorems, with the last stage serving as the final correctness check.

6. Relation to classical Ramsey-goodness and neighboring saturation notions

The adjective “Ramsey-good” is used in two distinct literatures. In the classical Burr–Erdős framework, for graphs $R(s,t)$13 and $R(s,t)$14, if $R(s,t)$15 is connected then

$R(s,t)$16

and $R(s,t)$17 is called $R(s,t)$18-good when equality holds (Pokrovskiy et al., 2015, Hu et al., 2022, Balla et al., 2016). In that sense, results such as $R(s,t)$19 being $R(s,t)$20-good for all $R(s,t)$21, large bounded-degree trees being $R(s,t)$22-good, and every tree or forest being good against $R(s,t)$23 belong to an exact-Ramsey-number theory rather than to the $R(s,t)$24-good extremal theory of doubly saturated graphs (Pokrovskiy et al., 2015, Balla et al., 2016, Hu et al., 2022).

This distinction matters because doubly saturated $R(s,t)$25-good graphs are not defined through the equality

$R(s,t)$26

but through the simultaneous inequalities $R(s,t)$27 and $R(s,t)$28, together with edge-maximality and edge-minimality inside that class (Przybocki et al., 23 Apr 2026). The 2022 work on trees and forests versus disconnected clique unions is thematically close to “double” obstruction on the blue side, but it explicitly does not use the phrase “doubly saturated” (Hu et al., 2022).

A second neighboring notion comes from saturation with respect to Ramsey-minimal families. A graph $R(s,t)$29 is $R(s,t)$30-saturated if it contains no Ramsey-minimal witness for $R(s,t)$31, but adding any missing edge creates one (Rolek et al., 2018). This is a different second-order extremal theory: it saturates with respect to a family of Ramsey-minimal graphs rather than requiring a single graph to be both edge-maximal and edge-minimal among $R(s,t)$32-good graphs.

Taken together, these distinctions place doubly saturated $R(s,t)$33-good graphs in a precise niche. They are “just barely” $R(s,t)$34-good, isolated under both edge insertion and edge deletion, historically rooted in circulant examples, and now supported by an explicit infinite family for $R(s,t)$35 together with a computational and formal-verification framework that appears likely to shape further progress (Przybocki et al., 23 Apr 2026).