extremal combinatorics

Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

\begin{problem} Bound the extremal number of $C_{10}$ in the hypercube. \end{problem}

Keywords: cycles; extremal combinatorics; hypercube

Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

Author(s): Morrison; Noel

\begin{problem} Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube. \end{problem}

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

\begin{question} Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$, then every saturated $k$-Sperner system $\mathcal{F}\subseteq \mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$? \end{question}

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

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