Status
On this page
Status
Topics
Status
On this page
Status
Topics
Given any distinct points in let count the number of distinct lines determined by these points. What are the possible values of ?
Given any distinct points in let count the number of distinct lines determined by these points. What are the possible values of , for all sufficiently large ?
Source: erdosproblems.com/606
An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.
Solved, in the site's label, which its commentary qualifies as holding for all sufficiently large , so the label describes the corrected Statement. The accepted full claim Salamon–Erdős 1988 determines the possible values for every , which answers the corrected Statement. The accepted partial claim Erdős 1972 determines the values above .
The site's wording asks for the possible values of for every , as Grünbaum's question is stated in Erdős's 1972 paper (p. 23) and in Salamon and Erdős [ErSa88]. The site labels the problem SOLVED and its commentary says "Solved (for all sufficiently large ) completely by Erdős and Salamon [ErSa88]; the full description is too complicated to be given here". The curator's reading is the question for all sufficiently large , and the corrected Statement adds those words; nothing else changes. The evidence for the discrepancy is the paper itself. Salamon and Erdős write (p. 137) that their formulas give "a complete answer to Grünbaum's problem for " but leave "the problem for open", that this case "requires a detailed analysis of the lower end of the high bands and appears to be difficult", and that "the size of is unknown but it is likely to be small"; their figure 5 shows, for , values at the lower end of the continuum below that the large- formulas omit. The answer under each reading: the corrected Statement is answered by [ErSa88], which determines the set of values of for every band by band, with Erdős's 1972 theorem supplying the values above and the paper fixing the best constant for the lower end of the continuum; the question for every , as printed, is open for , a threshold the paper does not compute, so no finite list of exceptional is on record. No result about the site's wording beyond these two papers is known here. [Er85], the site's source key, records Grünbaum's question of which values the number of lines can take, and reports without proof that every value above occurs except and (pp. 2-3, source page).