On some extremal problems on r-graphs

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Abstract

Denote by G(r)(n;k) an r-graph of n vertices and k r-tuples. Turán's classical problem states: Determine the smallest integer f{hook}(n;r,l) so that every G(r)(n;f{hook}(n;r,l)) contains a K(r)(l). Turán determined f{hook}(n;r,l) for r = 2, but nothing is known for r > 2. Put limn=- f{hook}(n;r,l) ( n r)= cr,l. The values of cr,l are not known for r > 2. I prove that to every e > 0 and integer t there is an n0 = n0(t,ε{lunate}) so that every G(r)(n;[(cr,l+ε{lunate})( n R)]) has lt vertices xt(j), l≤i≤t,l≤j≤l, so that all the r-tuples {Xi1(j1),...Xir(jr)}, l≤is≤t,l≤jl< ... <jr≤l, occur in our G(r). Several unsolved problems are posed.

Original languageEnglish
Pages (from-to)1-6
Number of pages6
JournalDiscrete Mathematics
Volume1
Issue number1
DOIs
Publication statusPublished - May 1971

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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