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Let  r  be an integer,  f ( n ) be a function, and  H  be a graph. Introduced by Erdős, Hajnal, Sós, and Szemerédi, the  r ‐Ramsey–Turán number of  H ,  RT   r  ( n ,  H ,  f ( n )), is defined to be the maximum number of edges in an  n ‐vertex,  H ‐free graph  G  with α  r  ( G )⩽ f ( n ), where α  r  ( G ) denotes the  K   r  ‐independence number of  G .  In this note, using isoperimetric properties of the high‐dimensional unit sphere, we construct graphs providing lower bounds for  RT   r  ( n ,  K   r + s  ,  o ( n )) for every 2⩽ s ⩽ r . These constructions are sharp for an infinite family of pairs of  r  and  s . The only previous sharp construction (for such values of  r  and  s ) was by Bollobás and Erdős for  r = s =2.

Some exact Ramsey–Turán numbers