Approximate form equivalent to Seymour’s conjecture

Establish that for every ε > 0, every oriented graph G has a vertex v such that |N^+_2(G,v)| ≥ (1−ε) |N^+_1(G,v)|.

Background

Using lexicographic product constructions, the paper shows that if Seymour’s conjecture fails then one can build digraphs in which all vertices violate the inequality by a fixed multiplicative margin. Conversely, the authors note an equivalent approximate formulation: proving that for every ε > 0 there is always a vertex with |N^+_2| ≥ (1−ε)|N^+_1| would settle Seymour’s conjecture.