Strong Dirac conjecture

Determine whether there exists a constant c > 0 such that every finite set X of n points in the plane, not all lying on a single line, contains an element incident with at least ⌈n/2⌉ − c lines spanned by X.

Background

The paper distinguishes the original Dirac–Motzkin conjecture from a weaker asymptotic statement. The original bound is known to have counterexamples, including infinite families, whereas the weak Dirac conjecture establishes only a linear lower bound with an unspecified positive constant.

Brass, Moser, and Pach proposed the strong Dirac conjecture as an asymptotic version retaining the sharp leading term ⌈n/2⌉ up to an additive constant. The paper does not resolve this conjecture; its main theorem instead proves the exact ⌈n/2⌉ bound for point sets distributed on three concurrent lines.