General Edge-isoperimetric Inequalities, Part II: a Local-Global Principle for Lexicographical Solutions

The lexicographical order ᏸ on a sequence space ᐄ n ϭ ͕ 0 , 1 ,. .. , ␣ ͖ n , defined by x n Ͻ ᏸ y n if f there exists a t such that x t Ͻ y t and x s ϭ y s for s Ͻ t , is one of the most important and frequently encountered orders in combinatorial extremal theory. An early result in this area , Harper's solution of an edge-isoperimetric problem (EIP) in binary Hamming space ([13]) (generalized in [16] to non-binary cases and rediscovered many times ; see , e. g , [6] , [9] and [15]) says that first segments in ᏸ are optimal. There are two kinds of EIP. They can be represented as extremal problems in graph theory. Let G ϭ (ᐂ , Ᏹ) be a graph. For any A ' ᐂ , define the set Ꮾ (A) of all boundary edges , that is , Ꮾ (A) ϭ ͕͕ x , y ͖ ෈ Ᏹ : ͉ ͕ x , y ͖ ʝ A ͉ ϭ 1 ͖ (1. 1) and the set ᏸ (A) of all inner edges ; that is , Ᏽ (A) ϭ ͕͕ x , y ͖ ෈ Ᏹ : x , y ෈ A ͖ .