On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity “within a given type”.

💡 Research Summary

The paper introduces a novel combinatorial model built on an infinite “difference graph” G whose vertex set is the natural numbers ℕ and where two vertices i and j are adjacent precisely when the absolute difference |i−j| belongs to a prescribed set D⊆ℕ⁺. This construction turns the adjacency relation into a purely arithmetic condition, allowing the authors to study permutation families that are forced to differ on at least one pair of adjacent vertices of G.

A central definition is that of a G‑different pair of permutations. For σ,τ∈Sₙ (the symmetric group on