A Pseudoline Counterexample to the Strong Dirac Conjecture

The Strong Dirac conjecture, open in some form since 1951 [5], is that every set of n points in R includes a member incident to at least n/2 − c lines spanned by the set, for some universal constant c. The less frequently stated dual of this conjecture is that every arrangement of n lines includes a line incident to at least n/2 − c vertices of the arrangement. It is known that an arrangement of n pseudolines includes a line incident to at least cdn vertices of the arrangement, for some universal constant cd [2, 8]. Every known infinite family of arrangements includes a line incident to at least n/2 − 3/2 vertices of the arrangement, and such a family was found by Felsner [4, p. 313]. I presented an infinite family of arrangements of n pseudolines that does not include any pseudoline incident to more than 4n/9− 10/9 vertices. Felsner found an infinite family of arrangements with n = 12k + 7 lines, each line incident to at most n/2 − 3/2 vertices. The first member of this family is shown in Figure 1.