Shadows of ordered graphs

  • Authors:

  • Béla Bollobás;Graham Brightwell;Robert Morris

  • Affiliations:

  • Trinity College, Cambridge CB2 1TQ, England, United Kingdom and Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, United States;Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, England, United Kingdom;IMPA, Estrada Dona Castorina 110, Jardim Botínico, Rio de Janeiro, RJ, Brazil

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

Isoperimetric inequalities have been studied since antiquity, and in recent decades they have been studied extensively on discrete objects, such as the hypercube. An important special case of this problem involves bounding the size of the shadow of a set system, and the basic question was solved by Kruskal (in 1963) and Katona (in 1968). In this paper we introduce the concept of the shadow @?G of a collection G of ordered graphs, and prove the following, simple-sounding statement: if n@?N is sufficiently large, |V(G)|=n for each G@?G, and |G|=|G|. As a consequence, we substantially strengthen a result of Balogh, Bollobas and Morris on hereditary properties of ordered graphs: we show that if P is such a property, and |P"k|