Seven-edge transversal bounds

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source_notes/: Paper summaries and source comparisons used in the research on Problem 644.

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Where things stand

Neither assertion of Problem 644 has been resolved. In the asymptotic range the unrestricted bounds remain

$$\lceil3k/4\rceil\le f(k,7)\le\lceil7k/8\rceil.$$

The bounds are due to Fon-der-Flaass, Kostochka, and Woodall. No fixed-positive-excess counterexample or unrestricted convergence theorem is known.

Source caution

The supplied 1997 transcription's floor roundings for both $r=4$ and $r=5$ conflict with nonvacuous examples: all triples on seven points have $P_4$ and transversal number five, while all triples on six points have $P_5$ and transversal number four. These bounded discrepancies do not challenge the reported asymptotic coefficients; no argument here relies on the disputed exact formulas.