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Register a new userA Note on the Frankl Conjecture
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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 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 $delta_r>0$.
The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Furedi in cite{FF} conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-graphs with $m$ edges. In this paper, we give some partial results for this conjecture.
Let $G=(V(G), E(G))$ be a multigraph with maximum degree $Delta(G)$, chromatic index $chi(G)$ and total chromatic number $chi(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $chi(G)leq Delta(G)+mu(G) +1$ for a multigraph $G$, where $mu(G)$ is the multiplicity of $G$. Moreover, Goldberg conjectured that $chi(G)=chi(G)$ if $chi(G)geq Delta(G)+3$ and noticed the conjecture holds when $G$ is an edge-chromatic critical graph. By assuming the Goldberg-Seymour conjecture, we show that $chi(G)=chi(G)$ if $chi(G)geq max{ Delta(G)+2, |V(G)|+1}$ in this note. Consequently, $chi(G) = chi(G)$ if $chi(G) ge Delta(G) +2$ and $G$ has a spanning edge-chromatic critical subgraph.
In this note we consider submersions from compact manifolds, homotopy equivalent to the Eschenburg or Bazaikin spaces of positive curvature. We show that if the submersion is nontrivial, the dimension of the base is greater than the dimension of the fiber. Together with previous results, this proves the Petersen-Wilhelm Conjecture for all the known compact manifolds with positive curvature.
We use logarithmic {ell}-class groups to take a new view on Greenbergs conjecture about Iwasawa {ell}-invariants of a totally real number field K. By the way we recall and complete some classical results. Under Leopoldts conjecture, we prove that Greenbergs conjecture holds if and only if the logarithmic classes of K principalize in the cyclotomic Z{ell}-extensions of K. As an illustration of our approach, in the special case where the prime {ell} splits completely in K, we prove that the sufficient condition introduced by Gras just asserts the triviality of the logarithmic class group of K.Last, in the abelian case, we provide an explicit description of the circular class groups in connexion with the so-called weak conjecture.
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