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Stationary Common Neighbors and Partition Hypotheses
From an $ω_1$-Ramsey cardinal we force that every countable coloring of $[ω_2]^2$ has a stationary set $X$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. The resulting monochromatic graph has diameter at most two after any nonstationary
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