A Higher-Order Clique Density Theorem

Abstract: Reiher's clique density theorem determines the sharp lower envelope for the density of $K_r$ at fixed edge density. We prove a higher-order version in which the prescribed quantity is itself a clique density. For every $3\le s<r$, we determine the minimum possible $K_r$-density among graphons with prescribed $K_s$-density. For $s\ge3$ the constraint is genuinely nonlinear and leaves the edge density undetermined; nevertheless, on the positive range the sharp lower boundary is the classical multipartite edge-to-clique profile, reparametrised by $K_s$-density. We also prove stability on the positive branches of this profile: at every interior point, near extremality forces cut-distance closeness to the corresponding extremal family at the induced edge density.

• The paper establishes the sharp lower bound of Kᵣ density in graphons with a given Kₛ density using explicit multipartite profiles.
• It employs a variational method and link induction to reduce the complexity of higher-order extremal problems to classical edge-to-clique cases.
• Stability results confirm that near-extremal graphons and large graphs are structurally close to balanced multipartite configurations.

Higher-Order Clique Density Theorems: Extending Extremal Graph Theory

Background and Context

This paper, "A Higher-Order Clique Density Theorem" (2607.06545), significantly advances the quantitative theory of extremal combinatorics, specifically in the context of clique densities in graphs and graphons. Building on fundamental results such as Turán's theorem and its sharpened extensions—especially Reiher's resolution of the Lovász-Simonovits conjecture ([Reiher 2016])—this work addresses the next logical layer by investigating the minimum possible density of $K_r$ (the $r$-clique) in graphs and graph limits (graphons), where the specified parameter is now not edge density ($K_2$-density) but rather the density of a smaller clique $K_s$ for $3 \leq s < r$.

Historically, determining the minimum number of $r$-cliques in a graph with a given edge density has driven much of extremal combinatorics, but higher-order constraints—prescribing smaller clique densities—introduce substantial nonlinearity and complexity. The classical result for edge densities was established for graphs by Razborov, Nikiforov, and ultimately Reiher. This paper extends these results to higher-order settings, formalizing and resolving a series of previously open questions.

Main Results

Clique-to-Clique Density Lower Bound

The central theorem determines, for all $3 \leq s < r$, the sharp lower bound for $K_r$-density among all graphons with prescribed $K_s$-density. For the range $s \geq 3$, the $r$0-constraint is inherently nonlinear and can no longer be expressed solely in terms of edge densities. Despite this, the extremal configurations remain multipartite graphons, as in the classical theory, but now parameterized by $r$1-density rather than edge density. The precise lower envelope function $r$2 is constructed using the classical multipartite graphon profiles, but reparametrized in terms of $r$3 rather than $r$4.

Formally, for every symmetric measurable graphon $r$5, the $r$6-density is

$r$7

The main theorem states that for every $r$8 and every graphon $r$9,

$K_2$0

where $K_2$1 is the clique-to-clique multipartite profile, defined via explicit multipartite construction. Equality is achieved only for graphons corresponding (up to measure-preserving isomorphism) to appropriately balanced multipartite structures.

Nonlinear Constraint and Multipartite Profiles

For $K_2$2, this is exactly Reiher's theorem. For $K_2$3, the extension is not a straightforward marginalization because the $K_2$4-density constraint does not fix the edge density—allowing more freedom for how cliques are distributed. Nevertheless, the results show that there is no structural advantage to dispersing $K_2$5-cliques unevenly: the balanced multipartite construction remains extremal throughout the domain.

The paper introduces a detailed analytic and combinatorial description of the multipartite profiles and demonstrates the continuity and monotonicity properties essential for establishing the sharpness of the bound and the structure of minimizers.

Stability Results

In analogy with the classical triangle and clique density theorems, the paper proves stability: near-extremal graphons and graphs are close (in cut distance or edit distance, respectively) to the extremal multipartite forms. This covers both the full graphon space and the finite graph setting, including all branches (zero and positive ranges) of the multipartite profile.

The key stability theorems provide quantitative and structural guarantees:

• If a graphon nearly achieves the bound, then its structure is close in cut metric to an extremal multipartite graphon with the same prescribed $K_2$6-density.
• For finite large graphs, if the clique-to-clique density is within $K_2$7 of the lower bound, the graph can be converted to a member of the extremal family by changing at most $K_2$8 edges.

A central technical innovation is the reduction of higher-order extremality (in $K_2$9) to the classical (edge-to-clique) case, utilizing a variational method and analysis of the local structure in links of graphons, combined with induction on clique size.

Proof Techniques and Analytic Framework

The proof employs the dense graph limit framework (graphons), variational calculus, and inductive arguments:

• Variational Approach: The proof minimizes $K_s$0 and analyzes first variations of the graphon, leading to strong regularity properties for minimizers.
• Link Induction: Exploiting the structure of vertex links, the problem is reduced recursively to lower values of $K_s$1, leveraging Reiher's original theorem as the base case.
• Critical Values: The points where profile branches meet are carefully treated using an adjacent clique recursion and analytic estimates, ensuring continuity and differentiability factors are managed rigorously.
• Stability via Compactness: With the equality cases characterized, stability follows from compactness arguments and reliance on prior sharp stability results for the edge-to-clique density case.

Implications and Future Perspectives

This paper generalizes the landscape of clique density extremal problems, showing that sharp bounds and stability results extend predictably to higher-order constraints. The explicit lower envelope $K_s$2 encapsulates the transition from local to global density control for arbitrary clique sizes. This work closes a natural gap in the theory of extremal densities, reinforcing the primacy of multipartite configurations and providing the mathematical infrastructure for further extremal and stability questions involving subgraph densities.

From a theoretical point of view, the extension and stabilization of flag algebra methods and variational link arguments to genuinely nonlinear constraints strengthens the analytic toolkit in extremal graph theory. Practically, these results inform the analysis of complex networks where the density of higher-order motifs is of interest, potentially impacting problems in network science, statistical physics, and probabilistic combinatorics.

Future research may investigate analogous density extremal results for other graph classes (e.g., for cycles or other subgraphs), extensions to sparse graph limits, or even broader functional inequalities in graphon space where multiple motif densities are prescribed simultaneously.

Conclusion

"A Higher-Order Clique Density Theorem" establishes the exact minimum $K_s$3-density for prescribed $K_s$4-density in graphons, generalizing Reiher’s theorem and demonstrating the continued extremality and stability of multipartite configurations for all $K_s$5. This paper integrates sophisticated analytic and combinatorial methods to advance the quantitative extremal theory of dense graphs and graph limits (2607.06545).