8 problems

Let $U(n)$ be the maximum number of unordered pairs of points at Euclidean distance $1$ among any set of $n$ points in the plane, that is,…

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The sources identify Erdős problem $\#608$ as an extremal problem involving a claimed threshold of $2/9$, but do not provide the underlying objects or define the extremal quantity…

Let $G$ be a graph on $n$ vertices with more than $n^2/4$ edges. Then there exists a triangle $T$ in $G$ and other vertices $y_1,\ldots,y_t$, where $t>(\frac{1}{2}-o(1))n$, such th…

Let $R$ be an operator-only Pure Recursive Calculus with a recursor rule of the form … The step argument $s$ is unrestricted, and an internally definable measure means a measure sa…

Let $T_{1}$ be the base theory for list concatenation. For a finite sequence of pairwise distinct list variables $\vec{X}=(X_{1},\dots,X_{m})$ and a sequence of non-zero natural nu…

In three-dimensional space, consider packings of congruent non-overlapping spheres, and let their density be the proportion of space occupied by the spheres, interpreted as the asy…