3 research outputs found

    On a problem of Henning and Yeo about the transversal number of uniform linear systems whose 2-packing number is fixed

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    For r2r\geq2, let (P,L)(P,\mathcal{L}) be an rr-uniform linear system. The transversal number τ(P,L)\tau(P,\mathcal{L}) of (P,L)(P,\mathcal{L}) is the minimum number of points that intersect every line of (P,L)(P,\mathcal{L}). The 2-packing number ν2(P,L)\nu_2(P,\mathcal{L}) of (P,L)(P,\mathcal{L}) is the maximum number of lines such that the intersection of any three of them is empty. In [Discrete Math. 313 (2013), 959--966] Henning and Yeo posed the following question: Is it true that if (P,L)(P,\mathcal{L}) is a rr-uniform linear system then τ(P,L)P+Lr+1\tau(P,\mathcal{L})\leq\displaystyle\frac{|P|+|\mathcal{L}|}{r+1} holds for all k2k\geq2?. In this paper, some results about of rr-uniform linear systems whose 2-packing number is fixed which satisfies the inequality are given
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