A density Corrádi-Hajnal Theorem

Abstract: We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles. This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corradi-Hajnal Theorem.

• The paper establishes the maximum edge count in n-vertex graphs avoiding k+1 vertex-disjoint triangles via four specific, extremal graph constructions.
• It employs a refined six-part vertex decomposition and intricate rotation arguments to control local structure and ensure stability.
• The explicit phase transitions between graph families reveal sharp density thresholds, bridging classical Corrádi-Hajnal results with modern density methods.

A Density Corrádi-Hajnal Theorem: An Expert Analysis

Introduction and Context

The manuscript "A density Corrádi-Hajnal Theorem" (1403.3837) addresses a classical extremal problem in graph theory: determining the maximal edge count in an $n$-vertex graph that avoids $k+1$ vertex-disjoint triangles. This problem generalizes Mantel’s theorem and incorporates aspects of the Corrádi-Hajnal theorem into a density-centric framework. The authors systematically characterize extremal graphs for all $k$ and sufficiently large $n$, providing a comprehensive answer to a longstanding question with deep connections to matching theory, tiling extremal problems, and the broader Turán-type context.

Main Result and Methodology

The principal theorem determines, for each $k$ and large $n$, the precise maximum number of edges in graphs not containing $k+1$ vertex-disjoint $K_3$s (triangles). For this, the extremal construction transitions between four graph families, $E_1(n, k)$ through $E_4(n, k)$, each optimized for a specific range of $k$:

• $E_1(n, k)$: Joins $k$ vertices (forming a complete graph) to a (nearly) balanced complete bipartite graph, optimized for small $k$.
• $E_2(n, k)$: Centers around a $(2k+1)$-vertex clique joined to a large independent set and between-part edges, becoming dominant as $k$ grows.
• $E_3(n, k)$: A structure where a large set $X$ is complete to its complement, relevant for intermediate $k$.
• $E_4(n, k)$: For large $k$, a more complex multipartite construction, degenerating to $K_n$ as $k \to n/3$.

The maximal edge count aligns with the maximum across these constructions: $e(G) \leq \max_{j \in [4]} e(E_j(n, k)).$

The proof employs a decomposition of the vertex set into six canonical parts (triangles of various interaction types, a maximum matching in the complement, and an independent set), using intricate rotation/augmentation arguments to control local structure. The authors invoke stability and counting results for triangle-intersecting sets and deploy a sequence of combinatorial optimizations to show that the extremal configurations must always mirror one of the four families.

A substantial technical component is verifying upper bounds through a detailed function analysis, tracking edge contributions between parts and ruling out the possibility of improving rotations—ultimately reinforcing the uniqueness and necessity of the extremal configurations.

Key Technical Features and Combinatorial Innovations

• Fine-Grained Decompositions: The six-part split (including refined notions of ‘types’ of triangles, matchings, and independent vertices) enables precise control over edge distribution, permitting upper bounds sharp enough to match the lower bound witness constructions.
• Rotation Methods and Stability: Through advanced rotation arguments (including local and large/complex configurations), the authors prove strong stability properties, ruling out non-extremal arrangements and ensuring all maximal configurations must structurally align with one of the $E_i(n, k)$.
• Density vs. Degree Thresholds: The result can be viewed as a density version of Corrádi-Hajnal, where the focus is on the total edge count rather than minimum degree. This perspective connects the result to more general density-based structures and the rich history of tiling and packings in extremal combinatorics.
• Asymptotic Sharpness and Explicit Transition Thresholds: For each $k$, explicit thresholds are given for transition points where one extremal construction yields to another, with matching achieved via careful calculations. This underscores the sensitivity of the extremal problem to the precise alignment of $n$ and $k$.

Numerical Features and Structural Phase Transitions

The main formula for the maximum number of edges is a piecewise function (the maximum of four explicit algebraic functions), with transition points delineated analytically:

• For $1 \leq k \leq (2n-6)/9$: $E_1(n,k)$ is extremal.
• For $(2n-6)/9 < k \leq (n-1)/4$: $E_2(n,k)$.
• For $(n-1)/4 < k \leq (5n-12+\sqrt{3n^2-10n+12})/22$: $E_3(n,k)$.
• For larger $k$, up to $n/3$, $E_4(n, k)$ dominates.

Transitions between these regimes are not continuous: shifting from one family to another requires $\Theta(n^2)$ modifications, illustrating the threshold phenomena and highlighting the rigidity of the extremal structure.

Implications and Theoretical Impact

This work provides an explicit, comprehensive description of the extremal graphs for forbidden $K_3$-tilings across all densities, filling a gap left by previous results that treated only sparse or very dense regimes, or gave asymptotic rather than exact answers. The clarity of the phase transitions and the match with prior degree-oriented theorems (Corrádi-Hajnal) solidifies the density framework as a robust setting for tiling extremal problems.

The approach generalizes the methods of matching theory and demonstrates that density extremal problems often require considering several non-equivalent families of constructions, with sharp structural thresholds. The methodological innovations in decomposition and rotation could be extended to analogous problems for larger cliques, hypergraphs, or more general tilings, and may inform progress on conjectures such as the Erdős Matching Conjecture or density versions of the Hajnal-Szemerédi theorem.

Connections to Broader Themes and Future Directions

This theorem is a model for density-based tiling questions in extremal combinatorics. The authors remark on possible generalizations to forbidden $K_r$-tilings, noting both the feasibility of the decomposition-rotation approach and the increased complexity as $r$ grows. Analogs for three-colorable graphs and further extensions to hypergraphs remain natural open directions.

The explicit stability results and reliance on triangle stability theorems (Erdős-Simonovits type) suggest further potential for analyzing maximal families not only from an existence perspective but also in terms of structural proximity.

From a practical perspective, the result characterizes the sharp edge-density threshold for robust triangle-packing, relevant to design theory, network reliability, and combinatorial optimization.

Conclusion

"A density Corrádi-Hajnal Theorem" supplies the exact extremal function for avoiding large collections of vertex-disjoint triangles in graphs of given size and edge-density. The intricate combinatorial analysis, coupled with explicit formulae and transition thresholds, elevates this result to a definitive statement in extremal graph theory, fusing classical matching/tiling phenomena with modern density-oriented techniques. The tools and concepts introduced herein are poised to influence future work in the field, particularly for density versions of broader tiling and partition problems.