Leader, Imre

Characterizing (aleph_0,aleph_1)-graphs ★★★

Author(s): Diestel; Leader

Call a graph an $(\aleph_0,\aleph_1)$-\emph{graph} if it has a bipartition $(A,B)$ so that every vertex in $A$ has degree $\aleph_0$ and every vertex in $B$ has degree $\aleph_1$.

\begin{problem} Characterize the $(\aleph_0,\aleph_1)$-graphs. \end{problem}

Keywords: binary tree; infinite graph; normal spanning tree; set theory

Few subsequence sums in Z_n x Z_n ★★

Author(s): Bollobas; Leader

\begin{conjecture} For every $0 \le t \le n-1$, the sequence in ${\mathbb Z}_n^2$ consisting of $n-1$ copes of $(1,0)$ and $t$ copies of $(0,1)$ has the fewest number of distinct subsequence sums over all zero-free sequences from ${\mathbb Z}_n^2$ of length $n-1+t$. \end{conjecture}

Keywords: subsequence sum; zero sum

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