On clique-to-clique densities

Abstract: Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\cdot)$ be the Lovász--Simonovits $r$-clique density function. For any integers $2\le s&lt;t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an $n$-vertex graph $G$ with a prescribed number $k_s(G)$, by showing that [ \frac{k_t(G)}{n^t}\ge F_t!\left(F_s^{-1}!\left(\frac{k_s(G)}{n^s}\right)\right), ] where $F_s<sup>{-1}$ denotes the generalized inverse. This strengthens Bollobás's piecewise-linear interpolation bound and, in the case $s=2$, recovers Reiher's clique density theorem via a new inductive proof.

• The paper establishes an asymptotically exact lower bound for t-clique density in terms of s-clique density for all 2 ≤ s < t.
• It leverages a novel inductive method on weighted graphs using the clique lifting theorem, Lagrange multipliers, and analytic envelope extensions.
• The results reveal structural rigidity and stability in extremal graphs, unifying classical Turán-type results with modern extremal approaches.

Asymptotically Sharp Bounds in Clique-to-Clique Density Problems

Background and Context

The study of clique densities, $k_r(G)$—the number of $r$-cliques in a finite graph $G$—is central to extremal graph theory. Classical results such as Turán's theorem and its generalization by Zykov and Erdős assert that the Turán graph $T_{r-1}(n)$ maximizes the number of smaller cliques within $K_r$-free graphs, and moreover, realize the maximal $k_s(G)$ for $K_r$-free graphs. This naturally leads to supersaturation and density problems: for given $k_s(G)$, what is the minimum attainable $k_t(G)$ for $t > s$? This is the $r$0 problem.

While analytic bounds and interpolations, especially for $r$1, have been progressively sharpened—from convex lower bounds (Khadžiivanov-Nikiforov), through piecewise-linear interpolations (Bollobás), to the final asymptotic sharpness of the clique density function $r$2 (Lovász–Simonovits conjecture, settled for all $r$3 by Reiher)—the multi-clique generality ($r$4 for all $r$5) remained incompletely resolved. The paper "On clique-to-clique densities" (2606.31967) closes this general case, providing sharp asymptotic lower bounds for $r$6 given $r$7, improving upon previous interpolation-style results and introducing new methodological insights.

Main Contributions

The principal result establishes the asymptotically exact lower bound for the $r$8-clique density in terms of the $r$9-clique density for arbitrary $G$0. Explicitly, for an $G$1-vertex graph $G$2, the normalized $G$3-clique count satisfies

$G$4

where $G$5 is the Lovász–Simonovits $G$6-clique density function, and $G$7 is its inverse on the strictly increasing domain.

This bound is shown to be:

• Sharp for all $G$8-clique densities, with extremality witnessed by multipartite constructions—specifically, the complete $G$9-partite graph with precisely optimized class sizes.
• A strict strengthening of Bollobás's piecewise-linear interpolation at all non-critical densities, highlighting the non-linearity and concavity properties of the supported region.
• A unified generalization: For $T_{r-1}(n)$0, the Reiher clique density theorem is recovered; for $T_{r-1}(n)$1, Turán-type extremal numbers emerge.

A corollary is a monotonic hierarchy of clique densities: for all $T_{r-1}(n)$2, $T_{r-1}(n)$3 forms an ascending sequence, each sharp at some graph.

Methodology and Proof Techniques

The approach is rooted in analytic and variational methods on the weighted graph model (Nikiforov), extending the proof architecture from previous extremal results. The key technical tool is the introduction of the clique lifting theorem: For weighted graphs, the $T_{r-1}(n)$4-clique density is lower-bounded in terms of the $T_{r-1}(n)$5-clique density through a function $T_{r-1}(n)$6, inductively propagating the lower bounds in a sharp fashion.

The proof differs from previous double induction (Reiher) by executing an induction on clique size $T_{r-1}(n)$7 alone. At each step:

• Weighted neighborhood graphs are employed; rooted clique densities and local analyses yield recursive lower bounds.
• Analytic envelope extensions and the study of concavity properties of the extended $T_{r-1}(n)$8 functions facilitate control over variations arising from local modifications.
• Lagrange multipliers are utilized to manage the constrained optimization inherent in the extremal problem.
• The proof of sharpness leverages the exact structure of multipartite extremal witnesses.

Auxiliary lemmas, such as quadratic clique-counting inequalities and monotonicity of the relevant density transformations, are essential for bounding clique numbers recursively.

Implications and Stability

The results have direct implications for extremal combinatorics:

• Structural Rigidity: The result identifies not only the minimum density configuration but shows that, up to $T_{r-1}(n)$9-edit distance, essentially only the constructed graphs can be extremal. Stability versions (under small density defect) are established, building on recent work on the asymptotic structure of triangle and general clique minimizers.
• Concavity and Generalization: The function $K_r$0 is shown to be piecewise concave, and the paper formulates a general problem of characterizing graph pairs $K_r$1 where this minimal density function is piecewise concave.
• Unified Extremal Framework: This completes the characterization for all clique-to-clique (and "lifted" clique) density relations in asymptotic regimes, providing a toolkit and inductive proof system that could extend to related problems in graph flag algebras and extremal flag enumeration.

Future Research Directions

Potential future developments follow naturally from the formulation of general extremal functions $K_r$2 and their concavity properties. In particular:

• General Graph Pairs: Determining, for arbitrary $K_r$3, whether the minimal $K_r$4-density given $K_r$5-density is piecewise concave, and characterizing extremal graphs when it is not.
• Refined Stability and Exact Results: While asymptotic stability is obtained, exact (finite $K_r$6) versions and enumeration for subsets of structurally constrained graphs remain open.
• Extension to Hypergraphs and Other Structures: Adapting the analytic and variational techniques to multigraphs, hypergraphs, or limits (graphons) could further generalize extremal combinatorial settings.

Conclusion

"On clique-to-clique densities" (2606.31967) sets the sharp asymptotic lower bound for the number of $K_r$7-cliques in an $K_r$8-vertex graph $K_r$9 with prescribed $k_s(G)$0-clique density, for all $k_s(G)$1. This resolves the generalized clique-to-clique supersaturation problem, supersedes previous piecewise-linear bounds, and unifies the extremal landscape for clique densities. The novel inductive strategy on weighted graphs not only advances the theoretical understanding of clique configurations but also establishes the stability of extremal structures and motivates further exploration of density extremal questions in combinatorics.