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تسجيل مستخدم جديدA Note on the Frankl Conjecture
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تأليف
Maysam Maysami Sadr
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The Frankl conjecture (called also union-closed sets conjecture) is one of the famous unsolved conjectures in combinatorics of finite sets. In this short note, we introduce and to some extent justify some variants of the Frankl 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 princi
pal 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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