Improved Crossing Lemma constants for other crossing-number variants

Improve the constant in the Crossing Lemma for the crossing-number, pair-crossing-number, and odd-crossing-number variants in Table 1 other than the versions governed by Rule +.

Background

The paper proves the basic Crossing Lemma with constant 1/64 for all nine combinations of crossing-number type and rules governing crossings between adjacent edges. However, the known improvements to the constant apply only to the Rule + variants.

The unresolved task is therefore to obtain sharper constants for the remaining variants, including the versions that allow or disregard crossings between adjacent edges according to Rules 0 and −.