7 problems

Let ${\cal G}_\kappa$ be a graph of cardinality $\kappa$, and let ${\cal T}_\kappa$ be the tree in which every vertex has degree $\kappa$. For a cardinal $\lambda$, the relation ……

For a coloring of unordered pairs of $\omega_1$ with two colors, write $\omega_1\rightarrow(\omega_1;\omega_1)^2_2$ for the stated polarized partition relation. Galvin's partition…

Let $complement$ be a partition of $omega$ into finitely many pieces. Graham–Rothschild's conjecture. One of the pieces contains all distinct sums from some infinite subset of…

Let $(n_i)_{i<\omega}$ be a sequence of cardinals, and let the bracketed condition $[n_i\geq i]$ have the meaning assigned by the source. Shelah's conjecture. … This is listed amon…

For sequences of cardinals $m_i,n_i,k_i$ indexed by $i<\omega$, let $\\{(m_i,n_i,k_i):i<\omega\\}\rightarrow(\lambda,\mu,k)$ denote the partition relation used in the source. Shela…

Closure conjecture. If $\theta<\alpha$, then

Let $\mu$ range over singular cardinals, and write $\operatorname{Pr}_1(\mu^+,\mu^+,\mu^+,\operatorname{cf}(\mu))$ for the corresponding strong coloring principle. Failure conjectu…