An extremal set theoretical characterization of some steiner systems

Abstract

Letn, k, t be integers,n>k>t≧0, and letm(n, k, t) denote the maximum number of sets, in a family ofk-subsets of ann-set, no two of which intersect in exactlyt elements. The problem of determiningm(n, k, t) was raised by Erdős in 1975. In the present paper we prove that ifk≦2t+1 andk−t is a prime, thenm(n, k, t)≦( n t )( 2k-t-1 k )/( 2k-t-1 t ). Moreover, equality holds if and only if an (n, 2k−t−1,t)-Steiner system exists. The proof uses a linear algebraic approach.

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Frankl, P. An extremal set theoretical characterization of some steiner systems. Combinatorica 3, 193–199 (1983). https://doi.org/10.1007/BF02579293

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AMS subject classification (1980)

  • 05 C 65
  • 05 B 05