Discrepancy After Adding A Single Set

We show that the hereditary discrepancy of a hypergraph F on n points increases by a factor of at most O(log n) when one adds a new edge to F . Let X be a set of n points. We say that a hypergraph F on X has discrepancy h if h is the smallest integer satisfying the following: There is a coloring χ: X → {−1, +1} such that for every edge S ∈ F , |χ(S)| ≤ h, where we write χ(S) for ∑ x∈S χ(x). The hereditary discrepancy of F is the maximum discrepancy of any restriction of F to a subset Y ⊆ X. Discrepancy and hereditary discrepancy are important notions in combinatorics and discrete geometry; for more information we refer, e.g., to [1] or [2]. Throughout the note, the asymptotic notation is used under the assumption that n → ∞. All logarithms have natural base. We denote by disc(F) and herdisc(F) the discrepancy and the hereditary discrepancy of F , respectively. The following question is a folklore in discrepancy theory (as far as we could find out, it was first asked by V. Sós some years ago): Given a hypergraph F , is it true that the hereditary discrepancy of F increases by at most a constant factor if one adds a new edge to F? As far as we know, there is no published result on this problem, although a polynomial factor can be proved by various arguments. In this note, we prove that adding one edge increases the hereditary discrepancy of F by a multiplicative factor of at most O(log n). Theorem 1 Let X be an n-point set and let F ⊆ 2X satisfy herdisc(F) ≤ h. Then disc(F ∪ {X}) = O(h log n). The following consequence is immediate from the definition of hereditary discrepancy: Corollary 2 Let X be an n-point set and let F ⊆ 2X satisfy herdisc(F) ≤ h. Then for any subset X ′ of X, we have herdisc(F ∪ {X }) = O(h log n).