Complete subgraphs in multipartite graphs

Abstract

Turán’s Theorem states that every graph G of edge density ${{\left\| G \right\|} \mathord{\left/ {\vphantom {{\left\| G \right\|} {\left( {_2^{\left| G \right|} } \right) > }}} \right. \kern-\nulldelimiterspace} {\left( {_2^{\left| G \right|} } \right) > }}\frac{{k - 2}} {{k - 1}}$ contains a complete graph K ^k and describes the unique extremal graphs. We give a similar Theorem for λ-partite graphs. For large λ, we find the minimal edge density d ^k _λ , such that every λ-partite graph whose parts have pairwise edge density greater than d ^k _λ contains a K ^k. It turns out that $d_\ell ^k = \frac{{k - 2}} {{k - 1}}$ for large enough λ. We also describe the structure of the extremal graphs.