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Source. Problem 36, p. 103, of P. Erdős, Research problems, Period. Math. Hungar. 15 (1984), no. 1, 101--103, doi:10.1007/BF02109375. The edition read is named on the source card.

Statement

Context. The constant $c$ is the absolute constant in Beck's theorem as reported: $n$ points with property $P_{n-k}$, $2\le k\le n$, determine at least $ckn$ distinct lines.

The remark (p. 103). Erdős writes that the value of $c$ given by Beck seems too small, and, quoted: "It would be tempting to conjecture that $c = 1/6$." He states that $c\le1/6$ follows from Sylvester's result, citing Burr, Grünbaum and Sloane, and adds that perhaps the conjecture $c=1/6$ is too optimistic and that one should first look for a counterexample.

Proof pointer

None in the paper; the bound $c\le1/6$ is attributed to Sylvester's result without argument, and Sylvester's estimate is display (3) on conjecture_p101.

Read depth

Claims checked: the three sentences were read clause by clause on the page image of p. 103. Nothing here is independently reviewed.

Dependencies

• theorem_p102_beck.
• Burr, Grünbaum and Sloane, The orchard problem (card burr_1974_orchard_problem).

Bears on

• Problem 211: the remark concerns the best constant in the problem's bound, a sharper question than the problem asks. The note poses it, states without proof that $c\le1/6$ follows from Sylvester's result, and proves nothing.