Lean-verified lower bounds for the Shannon capacity of odd cycles

2026-07-31

AI Summary

This work addresses the long-standing open problem of establishing lower bounds on the Shannon capacity of small odd cycles. By integrating Gao’s iterative method with techniques introduced by Itty et al., we present the first complete formal verification of these lower bounds within the Lean theorem prover, ensuring both mathematical rigor and computational precision. Leveraging constructions of independent sets from graph theory together with iterative optimization algorithms, we obtain the strongest known lower bounds to date: notably, Θ(C₇) ≥ 3.2588…, along with improved bounds for odd cycles C₁₁ through C₂₃. These results substantially advance the theoretical understanding of the Shannon capacity for small odd-cycle graphs.

Abstract

We give new lower bounds for the Shannon capacities of small odd cycles: $Θ(C_7)\geq3.258805369885\ldots$, $Θ(C_{11})\geq5.294502522149\ldots$, $Θ(C_{13})\geq6.302455083464\ldots$, $Θ(C_{15})\geq7.301600534487\ldots$, $Θ(C_{19})\geq9.357192705918\ldots$, $Θ(C_{21})\geq10.342455853338\ldots$, and $Θ(C_{23})\geq11.328224257774\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.

Problem

• Shannon capacity
• odd cycles
• lower bounds
• graph theory
• information theory

Innovation

• Shannon capacity
• odd cycles
• formal verification
• Lean
• lower bounds