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تسجيل مستخدم جديدOn Hypergraph Lagrangians and Frankl-Furedis Conjecture
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Frankl and Furedi conjectured in 1989 that the maximum Lagrangian, denoted by $lambda_r(m)$, among all $r$-uniform hypergraphs of fixed size $m$ is achieved by the minimum hypergraph $C_{r,m}$ under the colexicographic order. We say $m$ in {em principal domain} if there exists an integer $t$ such that ${t-1choose r}leq mleq {tchoose r}-{t-2choose r-2}$. If $m$ is in the principal domain, then Frankl-Furedis conjecture has a very simple expression: $$lambda_r(m)=frac{1}{(t-1)^r}{t-1choose r}.$$ Many previous results are focusing on $r=3$. For $rgeq 4$, Tyomkyn in 2017 proved that Frankl-F{u}redis conjecture holds whenever ${t-1choose r} leq m leq {tchoose r} -{t-2choose r-2}- delta_rt^{r-2}$ for a constant $delta_r>0$. In this paper, we improve Tyomkyns result by showing Frankl-F{u}redis conjecture holds whenever ${t-1choose r} leq m leq {tchoose r} -{t-2choose r-2}- delta_rt^{r-frac{7}{3}}$ for a constant $delta_r>0$.
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