Originator: Dhruv Mubayi (presented by Kevin Milans - REGS 2011)
Definitions: In $\mathbb{R}^d$, a $k$-corner consists of $k+1$ distinct points $x, y_1, \ldots, y_k$ such that $y_1 - x, \ldots, y_k - x$ are pairwise orthogonal. We say that $x$ is the center of the $k$-corner and that $y_1, \ldots, y_k$ are the legs. A $k$-corner is axis aligned if $y_j - x$ has one non-zero coordinate for each $j$. Let $f_{k,d}(n)$ be the maximum size of a subset of $\{0,\ldots,n-1\}^d$ that contains no $k$-corner, and let $g_{k,d}(n)$ be the maximum size of a subset of $\{0,\ldots,n-1\}^d$ that contains no axis-aligned $k$-corner.
Background: In a talk at the 2011 Cumberland Conference, Dhruv Mubayi noted that $f_{2,2}(n) = g_{2,2}(n) = 2(n-1)$. He asked for the value of $f_{2,3}(n)$, which appears to be unknown.
Problem: Determine bounds on $f_{k,d}(n)$ and $g_{k,d}(n)$ for $d\ge 3$.
Conjecture: $g_{k,d}(n) = 3\binom{n}{2} + 1$.
Comments: REGS participants have shown that $3\binom{n}{2} + 1 \le g_{2,3}(n) \le \frac{3}{2}n^2 - \frac{1}{2}(n-1)$ and have verified the conjecture for $n \le 5$. Since $f_{k,d}(n) \le g_{k,d}(n)$, it follows that $f_{2,3}(n) \le \frac{3}{2}n^2$. See the results page for other preliminary bounds.
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