Relational Complexity in Mathematics

• Relational Complexity is a measure that quantifies the minimum number of independent relations required to reconstruct global configurations from local subtuple data.
• It is formulated through permutation groups, with methods including base invariants and logarithmic bounds for primitive groups, demonstrating both theoretical depth and algorithmic relevance.
• RC guides the study of ultrahomogeneity in model theory and informs relational reasoning benchmarks in AI, impacting applications in graph theory and algebraic inference.

Relational complexity (RC) is a family of closely related arity parameters that quantify how much local relational information is needed before global behavior becomes determined. In permutation group theory, RC is the least $k$ for which $k$-subtuple orbit data determines orbit equivalence of all longer tuples; in the model theory of relational structures, it is the minimal arity of invariant relations needed to make a structure ultrahomogeneous; and in recent work on relational reasoning benchmarks, it is the minimum number of independent entities or operands that must be simultaneously bound to apply a relation (Kelsey et al., 2021, Hartman et al., 2013, Fesser et al., 14 Apr 2026). Across these settings, RC measures a transition from partial compatibility to full reconstruction.

1. Permutation-group formulation

Let $G \leq \mathrm{Sym}(\Omega)$ be a permutation group of degree $n=|\Omega|$. A sequence $A=(\omega_1,\dots,\omega_k)$ of points in $\Omega$ is a base for $G$ if the pointwise stabiliser $G_{\omega_1,\dots,\omega_k}$ is trivial. The minimum length of a base is $b(G)$. A base is minimal if no proper subsequence is itself a base; the maximum size of a minimal base is denoted $B(G)$. A base $k$0 is irredundant if

$k$1

and the maximum size of an irredundant base is $k$2. The height $k$3 is the size of the largest set $k$4 such that stabilisers shrink on adjoining any new point. These invariants satisfy

$k$5

For integers $k$6, two $k$7-tuples $k$8 and $k$9 in $G \leq \mathrm{Sym}(\Omega)$0 are called $G \leq \mathrm{Sym}(\Omega)$1-subtuple-complete, written $G \leq \mathrm{Sym}(\Omega)$2, if for every subset of indices $G \leq \mathrm{Sym}(\Omega)$3 there exists $G \leq \mathrm{Sym}(\Omega)$4 with

$G \leq \mathrm{Sym}(\Omega)$5

The relational complexity $G \leq \mathrm{Sym}(\Omega)$6 is the smallest $G \leq \mathrm{Sym}(\Omega)$7 such that for all $G \leq \mathrm{Sym}(\Omega)$8 and all $G \leq \mathrm{Sym}(\Omega)$9, the implication

$n=|\Omega|$0

holds. Equivalently, $n=|\Omega|$1 is the smallest $n=|\Omega|$2 such that the $n=|\Omega|$3-orbits on $n=|\Omega|$4 separate all orbits on $n=|\Omega|$5 for every $n=|\Omega|$6. The standard comparison with base-type invariants is

$n=|\Omega|$7

(Kelsey et al., 2021).

This formulation makes RC an orbit-reconstruction invariant. It asks when consistency on all $n=|\Omega|$8-coordinate projections forces global compatibility. The scarcity of exact values is a recurring feature of the subject: several papers emphasize that very few precise values of relational complexity are known.

2. Logarithmic bounds for primitive groups

A central result for finite primitive groups is due to Kelsey and Roney-Dougal. If $n=|\Omega|$9 is a primitive subgroup of $A=(\omega_1,\dots,\omega_k)$0 that is not of large-base type, then

$A=(\omega_1,\dots,\omega_k)$1

and hence

$A=(\omega_1,\dots,\omega_k)$2

Here “large-base” refers to the exceptional product-action and $A=(\omega_1,\dots,\omega_k)$3-on-$A=(\omega_1,\dots,\omega_k)$4-sets families in Liebeck’s classification. The same work shows that the maximal size of a minimal base and the height are both at most $A=(\omega_1,\dots,\omega_k)$5, and that a base of size at most $A=(\omega_1,\dots,\omega_k)$6 can be computed in polynomial time (Kelsey et al., 2021).

The proof proceeds through the O’Nan–Scott classification. In the almost simple case, detailed linear-algebraic chain-length arguments handle actions of $A=(\omega_1,\dots,\omega_k)$7 and $A=(\omega_1,\dots,\omega_k)$8 on subspaces. In product-action type $A=(\omega_1,\dots,\omega_k)$9, wreath-product chain-length arguments reduce the problem to the underlying primitive action. The remaining primitive types are handled using earlier work of Gill–Lodá–Spiga. A key general lemma is that if $\Omega$0 is transitive then

$\Omega$1

and also

$\Omega$2

The logarithmic bound is significant because it is the first universal logarithmic bound on $\Omega$3 and $\Omega$4 in the non-large-base case, and is stated to be best possible up to the constant factor. It also has algorithmic consequences. The base-finding procedure repeatedly chooses a point that reduces the current stabiliser by at least a factor of $\Omega$5, using orbit sizes and Schreier–Sims, and terminates after at most $\Omega$6 steps.

Concrete families illustrate the scale of the bound. For $\Omega$7 acting on points, $\Omega$8, $\Omega$9, and the chain-length argument gives

$G$0

so $G$1. For $G$2 acting on $G$3-subspaces with $G$4, one has

$G$5

which, for large $G$6, lies well below $G$7 of the orbit size.

3. Exact values for linear groups on subspaces

Freedman, Kelsey, and Roney-Dougal determine exact relational complexities for broad classes of linear groups acting on projective $G$8-spaces, and give general bounds for finite semilinear groups and for $G$9-space actions. Let $G_{\omega_1,\dots,\omega_k}$0, let $G_{\omega_1,\dots,\omega_k}$1, and let $G_{\omega_1,\dots,\omega_k}$2 be almost simple with

$G_{\omega_1,\dots,\omega_k}$3

For $G_{\omega_1,\dots,\omega_k}$4, they prove:

$G_{\omega_1,\dots,\omega_k}$5

and, if $G_{\omega_1,\dots,\omega_k}$6,

$G_{\omega_1,\dots,\omega_k}$7

These are among the few exact formulas presently available for primitive almost simple actions (Freedman et al., 2023).

The proof strategy has two complementary parts. For upper bounds, tuples of length at least $G_{\omega_1,\dots,\omega_k}$8 are reduced to a normal form whose first $G_{\omega_1,\dots,\omega_k}$9 entries are the coordinate lines $b(G)$0 and whose remaining entries have support size $b(G)$1 or $b(G)$2; combinatorial arguments on supports then show that sufficiently strong local equivalence implies orbital equivalence. For lower bounds, explicit witness tuples $b(G)$3 are constructed such that $b(G)$4 and $b(G)$5 are $b(G)$6-equivalent but are not in the same $b(G)$7-orbit. In the determinant-sensitive case, the obstruction is that every small subtuple can be matched by a determinant-one matrix, but no single element of $b(G)$8 carries the full tuple $b(G)$9 to $B(G)$0.

For finite fields, let $B(G)$1, set $B(G)$2, and let $B(G)$3 be the number of distinct prime divisors of $B(G)$4. Then, if $B(G)$5 and $B(G)$6,

$B(G)$7

with stronger lower bounds

$B(G)$8

For $B(G)$9-spaces, with $k$00 and $k$01,

$k$02

Several small-parameter examples are explicit. One recovers Cherlin’s result $k$03, and hence $k$04. Theorem A also gives $k$05. GAP computations reported in the paper include

$k$06

A further corollary is that there are infinitely many primitive non-large-base groups whose height minus relational complexity grows unboundedly.

4. Diagonal-type primitive groups

A primitive group of diagonal type has socle $k$07 with $k$08 a nonabelian simple group and $k$09, acting on a set $k$10 of size $k$11. One concrete realization is

$k$12

with $k$13 and

$k$14

The action is primitive exactly when either $k$15 or the top $k$16-factor acts primitively on $k$17 (Huang et al., 15 May 2026).

For this family, Huang and Roney-Dougal prove three qualitative facts. First, if $k$18 is primitive of diagonal type then

$k$19

Thus no diagonal-type primitive group is binary, and none has relational complexity $k$20. The lower bound is proved by constructing two $k$21-tuples $k$22 that are $k$23-subtuple-complete but lie in different $k$24-orbits. For $k$25, the construction uses

$k$26

with $k$27 a generating pair of $k$28.

Second, the lower bound is sharp for infinitely many examples. For $k$29 with $k$30 and $k$31, earlier work gives

$k$32

Hence $k$33, and combining this with the general lower bound yields $k$34 for infinitely many primitive diagonal groups.

Third, relational complexity is unbounded in this family. For any $k$35, if $k$36 and $k$37, then

$k$38

The argument constructs two $k$39-tuples that agree on any choice of $k$40 coordinates via suitable stabiliser elements but belong to distinct orbits globally.

The same paper relates RC to greedy base algorithms. If $k$41 denotes the largest base returned by the greedy algorithm that successively chooses a point in a largest orbit of the current stabiliser, then every greedy base is irredundant, so $k$42. For diagonal-type groups the paper determines the size of every greedy base produced in this way and proves Cameron’s conjecture for the family by establishing

$k$43

5. Relational complexity of relational structures

In the structural setting, let $k$44 be a relational structure with fixed signature $k$45, and let $k$46 be its automorphism group. A $k$47-ary relation $k$48 is an invariant of $k$49 if it is preserved by every automorphism. Writing $k$50 for the collection of invariant relations of arity at most $k$51, one forms the expansion

$k$52

The relational complexity of $k$53 is

$k$54

with $k$55 if no finite $k$56 suffices. The related lift complexity $k$57 is the minimum possible maximal arity of additional relations in an ultrahomogeneous lift of $k$58, and always satisfies

$k$59

This notion was introduced as a measure of ultrahomogeneity of a relational structure and was explicitly motivated by the original group-theoretic definition (Hartman et al., 2013).

Several basic properties are immediate or elementary. Relational complexity and lift complexity are closed under taking complements and monotone under adding invariant relations. For finite $k$60,

$k$61

For disjoint unions of connected components, $k$62 is governed by the maximum of the component complexities together with a unary correction term.

Graphs provide a large class of examples. A graph $k$63 has $k$64 exactly when it is ultrahomogeneous. By Gardiner’s theorem, the finite homogeneous graphs are disjoint unions of cliques $k$65, their complements, $k$66, and the line graph $k$67. If $k$68, then there is a finite partition of the vertex set into parts so that each induced subgraph is homogeneous and each bipartite graph between two parts is a homogeneous $k$69-edge-coloured bipartite graph. The cycle $k$70 has $k$71 by adjoining a binary “red edge” relation between vertices at graph distance $k$72.

For complexity $k$73, the paper highlights three important families: metrically ultrahomogeneous graphs, finite trees, and cographs. At the opposite end, finite graphs of arbitrarily large relational complexity exist. Johnson graphs satisfy

$k$74

with equality for large $k$75, while Kneser graphs satisfy

$k$76

In particular, the Petersen graph has $k$77.

A particularly strong theorem concerns universal structures defined by forbidden homomorphisms. If $k$78 is a finite minimal family of finite connected structures and $k$79 is the canonical universal, $k$80-categorical, existentially complete structure for $k$81, then if the largest minimal $k$82-separating $k$83-cut in $k$84 has size $k$85,

$k$86

Examples include the universal bipartite graph, with $k$87 when $k$88 is a single odd cycle; cographs, with $k$89 when $k$90; and the universal $k$91 structure, with $k$92.

6. Task-agnostic RC in relational reasoning benchmarks

Recent machine-learning work uses the same term in a task-agnostic sense. In “Evaluating Relational Reasoning in LLMs with REL,” relational complexity is defined as the minimal number of independent sources of variation that must be bound in parallel to carry out an inference, equivalently the arity of the relation:

$k$93

The paper distinguishes this from operand complexity (OC), which measures the difficulty of identifying or manipulating each filler once the arity is fixed. Two tasks can therefore have the same RC but different OC (Fesser et al., 14 Apr 2026).

The REL framework operationalizes this idea in three domains. REL-A studies algebraic relational reasoning through RPMs and RPTs. REL-B studies biological reasoning through homoplasy detection in phylogenetic settings and epistatic structure inference using Walsh–Hadamard coefficients

$k$94

REL-C studies chemical reasoning through constitutional isomer classification, maximum common substructure, completion of partial isomer families, and global constraint-satisfaction tasks. One evaluation metric used for maximum common substructure is

$k$95

A central methodological claim is that RC provides a principled axis along which reasoning difficulty can be varied while controlling for confounders such as input size, vocabulary, surface representation, and distractor sampling. The empirical finding is that across all domains and evaluation regimes, performance degrades sharply and almost monotonically with increasing RC, even when other factors are held fixed. Representative examples reported in the paper include the collapse of GPT-5.2 accuracy on algebraic tasks A3 and A4 from above $k$96 at $k$97 to near chance $k$98 at $k$99, the fall in biological homoplasy-detection accuracy from $G \leq \mathrm{Sym}(\Omega)$00 at $G \leq \mathrm{Sym}(\Omega)$01 to $G \leq \mathrm{Sym}(\Omega)$02 at $G \leq \mathrm{Sym}(\Omega)$03, and the decrease in chemistry task completion from $G \leq \mathrm{Sym}(\Omega)$04 on C1 to $G \leq \mathrm{Sym}(\Omega)$05 on C2 and $G \leq \mathrm{Sym}(\Omega)$06 on C3.

The reported interventions are longer contexts, in-context examples, best-of-$G \leq \mathrm{Sym}(\Omega)$07, majority voting, structured prompts, and tool use such as RDKit. These yield only marginal gains on high-RC instances and do not eliminate the monotonic failure pattern. The paper therefore treats higher-arity relational binding as a distinct failure mode rather than a mere prompt-length or token-budget effect.

7. Open problems and unsettled points

Several major questions remain open across the literature. In primitive permutation group theory, no primitive example of relational complexity $G \leq \mathrm{Sym}(\Omega)$08 is known. Diagonal-type groups exclude $G \leq \mathrm{Sym}(\Omega)$09 altogether, and it remains open whether an almost simple primitive group can have $G \leq \mathrm{Sym}(\Omega)$10 (Huang et al., 15 May 2026). In the non-large-base setting, stated future directions include sharpening the constant $G \leq \mathrm{Sym}(\Omega)$11, classifying cases of equality, and exploring lower bounds for other families of almost simple groups (Kelsey et al., 2021).

For linear groups, a natural open problem is to determine relational complexity for other classical groups—unitary, symplectic, and orthogonal groups—in their subspace actions, and to refine the constant factors in the $G \leq \mathrm{Sym}(\Omega)$12-space bounds (Freedman et al., 2023). In the structural theory of graphs and relational structures, the growth of

$G \leq \mathrm{Sym}(\Omega)$13

and the classification of structures with $G \leq \mathrm{Sym}(\Omega)$14 or $G \leq \mathrm{Sym}(\Omega)$15 are explicit open problems (Hartman et al., 2013). In recent benchmark work, future directions include extending controlled-RC evaluations to graph algorithms, multi-agent planning, and clinical decision-making (Fesser et al., 14 Apr 2026).

A persistent source of confusion is that the term “relational complexity” is not attached to a single formal object across all fields. In finite permutation groups it is an orbit-determination invariant; in model theory it is the minimal arity of invariant relations needed for ultrahomogeneity; and in recent reasoning benchmarks it is an arity-of-binding parameter. The common thread is nonetheless stable: RC quantifies how much simultaneous relational coordination is required before partial local agreement can no longer diverge from the global configuration.