Quasi‐symmetric designs with fixed difference of block intersection numbers

Abstract

AbstractThe following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved. Let D be a quasi‐symmetric design with z = y − x and v ≥ 2k. If x ≥ 1 + z + z3 then λ < x + 1 + z + z3. Let D be a quasi‐symmetric design with intersection numbers x, y and y − x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2. Let D be a triangle free quasi‐symmetric design with z = y − x and v ≥ 2k, then x ≤ z + z2. For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z. There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 49–60, 2007