Extremality of the five K4 constructions

Prove that, for sufficiently large n and all integers k with 0 ≤ k ≤ n/4, the five graph constructions E₁(n,k), E₂(n,k), E₃(n,k), E₄(n,k), and E₅(n,k) are extremal for ex(n,(k+1)K₄), equivalently that ex(n,(k+1)K₄) = max_{i∈[5]}|Eᵢ(n,k)|.

Background

The paper determines the asymptotic value of ex(n,(k+1)K₄) up to an O(n) error term by identifying five candidate extremal construction classes E₁(n,k),…,E₅(n,k), with different classes governing different ranges of k. The exact extremal assertion would remove the O(n) uncertainty and establish that no other n-vertex graph containing at most k vertex-disjoint copies of K₄ has more edges than the densest member of these five classes.

The authors state that their current method can establish the conjecture in the ranges k ∈ [0,(20+√10)n/130] ∪ [n/5,n/4], but the full assertion remains unresolved for the intervening parameter range. The conjecture is therefore a concrete strengthening of the paper’s asymptotic density theorem.

References

We conjecture that these constructions are extremal, i.e. \mathrm{ex}(n,(k+1)K_{4}) = \max_{i\in [5]}|E_{i}(n,k)|, when n is large.

— Density Hajnal--Szemerédi theorem for cliques of size four  (2501.00801 - Hou et al., 1 Jan 2025) in Remark (i) following Theorem Mian–Hajnal–Szemerédi density K4, Section 1.1, and repeated at the end of the subsection on extremal constructions