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Sunil Chandran (Indian Institute of Science, Rajat Adak","submitted_at":"2025-04-03T17:51:50Z","abstract_excerpt":"Let $G$ be a simple graph with $n$ vertices and $m$ edges. According to Tur\\'{a}n's theorem, if $G$ is $K_{r+1}$-free, then $m \\leq |E(T(n, r))|,$ where $T(n, r)$ denotes the Tur\\'{a}n graph on $n$ vertices with a maximum clique of order $r$. A limitation of this statement is that it does not give an expression in terms of $n$ and $r$. A widely used version of Tur\\'{a}n's theorem states that for an $n$-vertex $K_{r+1}$-free graph, $m \\leq \\left\\lfloor \\frac{n^2(r-1)}{2r} \\right\\rfloor.$ Though this bound is often more convenient, it is not the same as the original statement. 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