G. Eric Moorhouse:

Open Problems

Finite projective planes

1. Must the order of a finite projective plane be a prime power?
2. Must planes of prime order be Desarguesian?
3. Find an analogue of Segre's Theorem for cubics, i.e. find a combinatorial characterization of cubic curves in finite Desarguesian planes.
4. Classify translation planes of order 32.
5. Classify semifield planes of order 64.
6. Is every finite partial linear space embeddable in some finite projective plane?
7. Prove that every Desarguesian semibiplane lifts to a unique (Desarguesian) plane. [Currently this is known only in the case of prime order. The general case should be do-able; I just haven't given it enough attention.]

Generalised quadrangles

1. Does there exist a GQ(s,t) with finite s>2 and infinite t ?
2. Find, if possible, a finite generalised quadrangle Q with an automorphism τ of order 2, such that the quotient Q/τ lifts to a quadrangle other than Q.

Loops

1. Find a simple Bol loop (other than a Moufang loop).
2. Finish the classification of Bol loops of small order. [The only orders less than 32 for which this remains open are 24 and 30. This problem should be quite do-able.]
   More loop problems

Ovoids of Polar Spaces

1. Do ovoids exist in O⁺(10,q) for some prime power q?
2. Do ovoids exist in O⁺(8,25)?
3. Do ovoids exist in O(7,q), other than the known examples (for which q is a power of 3)?

Chromatic numbers of graphs

1. Must equality hold in the upper bound χ(Γ₁ × Γ₂ × ... × Γ_k) ≤ min{χ(Γ₁), ..., χ(Γ_k)} ?
2. Find a subgraph of the Euclidean plane graph having chromatic number at least 5.
3. Is the chromatic number of the plane constructed over the field of complex numbers, finite? This graph has pairs of complex numbers as vertices, and the vertex (u,v) is adjacent to (z,w) whenever (z–u)²+(w–v)²=1.
   S.T. Hedetniemi lists more such problems

 Under Construction