5 problems

Let $k\geq 3$ be an integer. A $k$-uniform hypergraph is linear if it contains no $2$-cycle, and let $\alpha({\mathcal H})$ denote its independence number. Shattering-threshold con…

Mészáros–Rónyai conjecture. For every nonempty s-extremal family $\mathcal{F}\subseteq 2^{[n]}$, there exists $F_0\in\mathcal{F}$ such that $\mathcal{F}\setminus\{F_0\}$ is still s…

Frankl's conjecture. Let $k$ be a non-negative integer and $n\geqslant 2k$. If $\mathcal{F}\subseteq 2^{[n]}$ is an antichain with $\mathcal{F}\not\rightarrow(k+1,2^{k+1})$, then

Sublogarithmic-bound conjecture. If

Let $T_s$ and $T_{sh}$ denote the static and shattering transition temperatures, respectively. For a temperature $T$, say that the free energy landscape is $(E,q,r)$-shattered when…