Existence of affine planes of non-prime-power order

Determine whether affine planes exist for any order that is not a prime power.

Background

An affine plane of order t is a finite incidence structure with t² points and t²+t lines, together with t+1 parallel classes satisfying the usual incidence and intersection properties. The construction used in the paper requires an affine plane of order r−1 to produce an r-edge-coloring whose monochromatic components attain the relevant upper bound.

Affine planes are known to exist whenever their order is a prime power. The unresolved existence question matters because an affine plane of order r−1 would yield the Gyárfás-type extremal coloring for the corresponding number r of colors, potentially establishing the matching upper bound for additional values of r.

References

Affine planes of order $t$ are known to exist whenever $t$ is a prime power; it is open whether affine planes of any other order exist.

— Monochromatic components with many edges in random graphs  (2509.01766 - Fox et al., 1 Sep 2025) in Section 2, subsection “Upper bounds and Gyárfás's construction”

However, it is suspected that all finite affine planes have prime-power order p.\ 144, so we obtain no relaxation of the conditions to Corollary~\ref{cor:coding_inductive_step}.

— On exceptional cliques in matrix rings  (2608.24586 - Boutros et al., 25 Aug 2026) in Remark following Example 4.2, Section 6 (Cliques from coding theory)