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Source. The displayed inequality of Section 4, p. 116, unnumbered, of Paul Erdős and Peter Fishburn, A postscript on distances in convex n-gons, Discrete Comput. Geom. 11 (1994), 111--117, doi:10.1007/BF02573998, as named on the source card; labels and pages are the print's own.

Statement

Setting (p. 111). $f(n)$ is the minimum over all convex $n$-gons of the maximum over the vertices of the number of distinct distances from that vertex to the other vertices. Erdős's conjecture C2 (p. 111), that some vertex has at least $\lfloor n/2\rfloor$ different distances to the other vertices, says $f(n)=\lfloor n/2\rfloor$, the regular polygon attaining $\lfloor n/2\rfloor$.

Inequality (p. 116). The paper states that its theorem gives

$$f(n)\ge\lfloor(n+3)/3\rfloor\qquad\text{for } n\ge4,$$

calling it a tiny improvement on Moser's lower bound $f(n)\ge\lfloor(n+2)/3\rfloor$ for C2, recorded on p. 111 as the best lower bound known to the authors, and adds that this is a very long way from $\lfloor n/2\rfloor$.

In the corpus's words: for every $n\ge4$, every convex $n$-gon has a vertex with at least $\lfloor n/3\rfloor+1$ distinct distances to the other vertices. It follows from the Theorem of p. 112 because the $k$ vertices of a run from $x_0$ lie at $k$ different distances from $x_0$. The bound equals $\lfloor n/2\rfloor$ for $n=4,5,6,7,9$ and is smaller for $n=8$ and every $n\ge10$ (arithmetic done here, not in the paper).

Read depth. Claims checked: the statement and the definition of $f$ were read clause by clause on printed pages 111 and 116. Nothing here is independently reviewed.

Proof pointer

Immediate from the Theorem of p. 112: take a vertex with a run of length $\lfloor(n+3)/3\rfloor$; the distances along the run are strictly increasing, so they are distinct.

Dependencies

Within the paper: the Theorem of p. 112.

Bears on

• Problem 982: a lower bound for the problem's statement, which asks for $\lfloor n/2\rfloor$ distinct distances from some vertex of a convex $n$-gon. It meets $\lfloor n/2\rfloor$ exactly for $n=4,5,6,7,9$ and falls short for $n=8$ and every $n\ge10$; the paper calls the problem open (p. 111).