Matroids with No (q+2)-Point-Line Minors |
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Authors: | Joseph E. Bonin |
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Affiliation: | Department of Mathematics, The George Washington University, Washington, D.C. 20052 |
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Abstract: | It is known that a geometry with rankrand no minor isomorphic to the (q+2)-point line has at most (qr−1)/(q−1) points, with strictly fewer points ifr>3 andqis not a prime power. Forqnot a prime power andr>3, we show thatqr−1−1 is an upper bound. Forqa prime power andr>3, we show that any rank-rgeometry with at leastqr−1points and no (q+2)-point-line minor is representable overGF(q). We strengthen these bounds toqr−1−(qr−2−1)/(q−1)−1 andqr−1−(qr−2−1)/(q−1) respectively whenqis odd. We give an application to unique representability and a new proof of Tutte's theorem: A matroid is binary if and only if the 4-point line is not a minor. |
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