Frequency Flow Theory: An Entropy Approach to Sunflower-Free Families - Complete Proof of the Growth Rate Theorem

We present Frequency Flow Theory, an entropy-based framework that provides a complete proof of the Growth Rate Theorem for sunflower-free families.

MAIN RESULT: For k ≥ 3, the maximum size of a k-sunflower-free family on [n] satisfies:

    m(n,k) = Θ((k/(k-1))^n)

For k=3: m(n,3) = Θ((3/2)^n)

KEY CONTRIBUTIONS:

  • Entropy decomposition via Shannon's chain rule: log₂|F| = Σᵢ H(Xᵢ | ...)

  • Conditional entropy formula: H(Xᵢ | ...) = 2|Xᵢ|/|F|

  • Blocking constraint from sunflower obstruction

  • Key inequality: c/(1-c) < 1/(k-1) where c = log₂(k/(k-1))/2

  • Complete inductive proof

The growth rate k/(k-1) emerges from the overlap-blocking trade-off: overlap enables family growth but creates blocking constraints. The optimal balance determines the growth rate.

Version 2.0 - Complete proof (10 pages)

Version 1.0 - Initial conjecture and empirical verification