The transversal numbers of linear systems with given 2-packing numbers

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Abstract

A linear system is a pair (X,F) where F is a finite family of subsets on a ground set X and it satisfies that |AB|1 for every pair of distinct subsets A,BF. We study the relation between two parameters in linear systems: the transversal and the 2-packing numbers. Our main theorem shows that any linear system with 2-packing number equal to k has transversal number at most k, for k={2,3,4}, and for k=4 the equality is attained only for one special family of linear subsystems of the projective plane of order 4.

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Research supported by CONACyT-México under Project 166306 and PAPIIT-México under Project IN101912.

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