Abstract
The Kruskal-Katona theorem is a celebrated result of extremal combinatorics providing precise cardinality bounds on the ‘shadow’ of a family of finite sets: the family given by removing an element from each set of the original. We describe a formalisation of the Kruskal-Katona theorem in the Lean theorem prover, including a computable implementation of the shadow as well as standard inequalities about it, and a definition of the colexicographic ordering on finite sets. In addition, we apply these results to other classical combinatorial theorems: Sperner’s theorem on antichains and the Erdős-Ko-Rado theorem on intersecting families.
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Acknowledgement
We would like to give particular thanks to Imre Leader for his inspiring lecture series with demonstrations of these proofs, Yaël Dillies for their continuous and determined efforts to migrate the code here to mathlib, and the anonymous reviewers for their helpful feedback.
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Mehta, B. (2022). Formalising the Kruskal-Katona Theorem in Lean. In: Buzzard, K., Kutsia, T. (eds) Intelligent Computer Mathematics. CICM 2022. Lecture Notes in Computer Science(), vol 13467. Springer, Cham. https://doi.org/10.1007/978-3-031-16681-5_5
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