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The deductions and obstructions here are author-recorded by Plasma AI. Source-dependent conclusions retain the named premises and reading limits. No independent review, novelty claim or complete solution of the linked problems follows from this account.

The relevant questions are E0107, E0838.

Convex configurations: E0107 and E0838

The Holmsen--Mojarrad--Pach--Tardos Theorem 1.3, read at statement level, applies to pseudoconfigurations and hence to real planar point sets in general position:

$$\operatorname{ES}(k)\leq 2^{k+O(\sqrt{k\log k})}.$$

It does not give the exact value 2^(k-2)+1 in E0107. A finite order-type test at one k needs either all realizable types or an exactly realized counterexample. Nonstretchable pseudoline arrangements are not real counterexamples, and checking a finite set of k values has no asserted extension principle.

For E0838 the same theorem gives a quantitative consequence. Put M=ES(k). Each M-subset contains a convex k-subset, while any particular convex k-subset lies in exactly binom(n-k,M-k) M-subsets. Therefore the number of convex k-subsets is at least

$$\frac{\binom nM}{\binom{n-k}{M-k}} =\frac{\binom nk}{\binom Mk}\geq(n/M)^k \quad(n\geq M).$$

With natural logarithms and k=floor(log n/(2 log 2)), log M=(log 2)k+o(k), so

$$\liminf_{n\to\infty}\frac{\log f(n)}{(\log n)^2} \geq\frac1{4\log2}.$$

For the other direction the classical Erdős--Szekeres construction has 2^(k-2) points with no convex k-subset. Take k=ceil(log_2 n)+2 and retain any n of those points. All convex subsets have size less than k, so

$$f(n)\leq\sum_{j<k}\binom nj\leq k n^{k-1}, \qquad \limsup_{n\to\infty}\frac{\log f(n)}{(\log n)^2} \leq\frac1{\log2}.$$

The old double-counting account was also read in Erdős 1978, inequality (2). The displayed sharpening uses the inspected modern existence theorem as a premise; its proof was not audited. These bounds neither prove a limit nor identify its value. Summing O(log n) size classes changes the logarithm by only O(log log n), so that alone cannot repair the leading constant. The next test is a construction controlling all convex sizes or a lower bound using more structure than one guaranteed polygon.

Source interfaces

These source locators identify the uses in this account. Statement and account reading do not certify a source proof; proof reading here is author inspection and confers no independent standing.

• [[../library/discrete_geometry/erdos_1978_more_problems_elementary_geometry/_index|erdos 1978 more problems elementary geometry]]: Statement and double-counting proof account (account reading).
• [[../library/discrete_geometry/holmsen_2020_two_extensions_erdos_szekeres_problem/_index|holmsen 2020 two extensions erdos szekeres problem]]: Introduction, pseudoconfigurations and Theorem 1.3 (statement reading). Theorem 1.3 and the classical 2^(k-2)+1 lower-bound construction in the introduction (statement reading).