Large complete bipartite subgraphs in incidence graphs of points and hyperplanes

Roel Apfelbaum*, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We show that if the number / of incidences between m points and n planes in ℝ3 is sufficiently large, then the incidence graph (which connects points to their incident planes) contains a large complete bipartite subgraph involving r points and s planes, so that rs ≥ I2/mn-a(m + n), for some constant a > 0. This is shown to be almost tight in the worst case because there are examples of arbitrarily large sets of points and planes where the largest complete bipartite incidence subgraph records only I 2/mn - m+n/16 incidences. We also take some steps towards generalizing this result to higher dimensions.

Original languageEnglish
Pages (from-to)707-725
Number of pages19
JournalSIAM Journal on Discrete Mathematics
Issue number3
StatePublished - 2007


  • Hyperplanes
  • Incidence graph
  • Incidences


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