Oct 20, 2015 - the polarity graph Go Ï as follows. V (Go Ï) = P and p â¼ q if and only if p â Ï(q). That is, the neighborhood of a vertex p is t...

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arXiv:1509.06587v2 [math.CO] 20 Oct 2015

Michael Tait∗

Abstract A conjecture widely attributed to Neumann is that all finite non-desarguesian projective planes contain a Fano subplane. In this note, we show that any finite projective plane of even order which admits an orthogonal polarity contains a Fano subplane. The number of planes of order less than n previously known to contain a Fano subplane was O(log n), whereas the number of planes of order less than n that our theorem applies to is not bounded above by any polynomial in n. Mathematics Subject Classification: 05C99, 51A35, 51A45

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Introduction

A fundamental question in incidence geometry is about the subplane structure of projective planes. There are relatively few results concerning when a projective plane of order k is a subplane of a projective plane of order n. Neumann [9] found Fano subplanes in certain Hall planes, which led to the conjecture that every finite nondesarguesian plane contains P G(2, 2) as a subplane (this conjecture is widely attributed to Neumann, though it does not appear in her work). Johnson [7] and Fisher and Johnson [4] showed the existence of Fano subplanes in many translation planes. Petrak [10] showed that Figueroa planes contain P G(2, 2) and Caliskan and Petrak [3] showed that Figueroa planes of odd order contain P G(2, 3). Caliskan and Moorhouse [2] showed that all Hughes planes contain P G(2, 2) and that the Hughes plane of order q 2 contains P G(2, 3) if q ≡ 5 (mod 6). We prove the following. Theorem 1. Let Π be a finite projective plane of even order which admits an orthogonal polarity. Then Π contains a Fano subplane. Ganley [5] showed that a finite semifield plane admits an orthogonal polarity if and only if it can be coordinatized by a commutative semifield. A result of Kantor [8] implies that the number of nonisomorphic planes of order n a power of 2 that can be coordinatized by a commutative semifield is not bounded above by any polynomial in n. Thus, Theorem 1 applies to many projective planes. ∗

Department of Mathematics, University of California San Diego. [email protected]

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Proof of Theorem 1

The proof of Theorem 1 is graph theoretic, and we collect some definitions and results first. Let Π = (P, L, I) be a projective plane of order n. We write p ∈ l or say p is on l if (p, l) ∈ I. Let π be a polarity of Π. That is, π maps points to lines and lines to points, π 2 is the identity function, and π respects incidence. Then one may construct the polarity graph Goπ as follows. V (Goπ ) = P and p ∼ q if and only if p ∈ π(q). That is, the neighborhood of a vertex p is the line π(p) that p gets mapped to under the polarity. If p ∈ π(p), then p is an absolute point and the vertex p will have a loop on it. A polarity is orthogonal if exactly n + 1 points are absolute. We note that as neighborhoods in the graph represent lines in the geometry, each vertex in Goπ has exactly n + 1 neighbors (if v is an absolute point, it has exactly n neighbors other than itself). We provide proofs of the following preliminary observations for completeness. Lemma 1. Let Π be a projective plane with polarity π, and Goπ be the associated polarity graph. (a) For all u, v ∈ V (Goπ ), u and v have exactly 1 common neighbor. (b) Goπ is C4 free. (c) If u and v are two absolute points of Goπ , then u 6∼ v. (d) If v ∈ V (Goπ ), then the neighborhood of v induces a graph of maximum degree at most 1. (e) Let e = uv be an edge of Goπ such that neither u nor v is an absolute point. Then e lies in a unique triangle in Goπ . Proof. To prove (a), let u and v be an arbitrary pair of vertices in V (Goπ ). Because Π is a projective plane, π(u) and π(v) meet in a unique point. This point is the unique vertex in the intersection of the neighborhood of u and the neighborhood of v. (b) and (c) follow from (a). To prove (d), if there is a vertex of degree at least 2 in the graph induced by the neighborhood of v, then Goπ contains a 4-cycle, a contradiction by (b). Finally, let u ∼ v and neither u nor v an absolute point. Then by (a) there is a unique vertex w adjacent to both u and v. Now uvw is the purported triangle, proving (e). Proof of Theorem 1. We will now assume Π is a projective plane of even order n, that π is an orthogonal polarity, and that Goπ is the corresponding polarity graph (including loops). Since n is even and π is orthogonal, a classical theorem of Baer ([1], see also Theorem 12.6 in [6]) says that the n + 1 absolute points under π all lie on one line. Let a1 , . . . , an+1 be the set of absolute points and let l be the line containing them. Then there is some p ∈ P such that π(l) = p. This means that in Goπ , the neighborhood of p is exactly the set of points {a1 , . . . , an+1 }. For 1 ≤ i ≤ n + 1, let Ni be the neighborhood of ai . Then by Lemma 1.b, Ni ∩ Nj = ∅ if i 6= j. Further, counting gives that ! ! n+1 n+1 [ [ V (Goπ ) = p ∪ ai ∪ Ni . (1) i=1

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Figure 1: ER2o

Let ER2o be the graph on 7 points which is the polarity graph (with loops) of P G(2, 2) under the orthogonal polarity.

Lemma 2. If ER2o is a subgraph of Goπ , then Π contains a Fano subplane. Proof. Let v1 , . . . , v7 be the vertices of a subgraph ER2o of Goπ . Let li = π(vi ) for 1 ≤ i ≤ 7. Then the lines l1 , . . . , l7 in Π restricted to the points v1 , . . . , v7 form a point-line incidence structure, and one can check directly that it satisfies the axioms of a projective plane. Thus, it suffices to find ER2o in Goπ . To find ER2o it suffices to find distinct i, j, k such that there are vi ∈ Ni , vj ∈ Nj , and vk ∈ Nk where vi vj vk forms a triangle in Goπ , for then the points p, ai , aj , ak , vi , vj , vk yield the subgraph ER2o . Now note that for all i, and for v ∈ Ni , v has exactly n neighbors that are not absolute points. There are n + 1 choices for i and n − 1 choices for v ∈ Ni . As each edge is counted twice, this yields n(n − 1)(n + 1) 2 edges with neither end an absolute point. By Lemma 1.e, there are at least n3 − n 6 triangles in Goπ . By Lemma 1.c, there are no triangles incident with p, by Lemma 1.b, there are no triangles that have more than one vertex in Ni for any i, and by Lemma n 1.d there are at most b n−1 2 c = 2 − 1 triangles incident with ai for each i. Therefore, by (1), there are at least n n3 − n − (n + 1) −1 6 2 copies of ER2o in Goπ . This expression is positive for all even natural numbers n.

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Concluding Remarks

First, we note that the proof of Theorem 1 actually implies that there are Ω n3 copies of P G(2, 2) in any plane satisfying the hypotheses, and echoing Petrak [10], perhaps one could find subplanes of order 4 for n large enough. We also note that it is crucial in the proof that the absolute points form a line. When n is odd, the proof fails (as it must, since our proof does not detect if Π is desarguesian or not).

Acknowledgments The author would like to thank Gary Ebert and Eric Moorhouse for helpful comments.

References [1] Reinhold Baer. Projectivities with fixed points on every line of the plane. Bulletin of the American Mathematical Society, 52(4):273–286, 1946. [2] Cafer Caliskan and G Eric Moorhouse. Subplanes of order 3 in hughes planes. The Electronic Journal of Combinatorics, 18(P2):1, 2011. [3] Cafer Caliskan and Bryan Petrak. Subplanes of order 3 in figueroa planes. Finite Fields and Their Applications, 20:24–29, 2013. [4] J Chris Fisher and Norman L Johnson. Fano configurations in subregular planes. Note di Matematica, 28(2):69–98, 2010. [5] MJ Ganley. Polarities in translation planes. Geometriae Dedicata, 1(1):103–116, 1972. [6] Daniel R Hughes and Frederick Charles Piper. Springer, 1973.

Projective planes, volume 6.

[7] Norman L Johnson. Fano configurations in translation planes of large dimension. Note di Matematica, 27(1):21–38, 2009. [8] William M Kantor. Commutative semifields and symplectic spreads. Journal of Algebra, 270(1):96–114, 2003. [9] Hanna Neumann. On some finite non-desarguesian planes. Archiv der Mathematik, 6(1):36–40, 1954. [10] Bryan Petrak. Fano subplanes in finite figueroa planes. Journal of Geometry, 99(1-2):101–106, 2010.

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