I have a combinatorics problem that seems pretty general - I'd be surprised if the answer is not known. Unfortunately, I can't seem to solve it.

The question concerns the following situation: Suppose we have a set $L$ composed of $k$ disjoint, parallel half-lines $L_i$ in $\mathbb{R}^3$ (drawn in blue below). Suppose we have another collection of $n$ pairwise-disjoint paths $P_j$ in $\mathbb{R}^3$ (drawn in the other colors with $n = 4$) so that if $P_j = \gamma_j([0, \infty))$ then $\gamma_j(0) \in L$, $\gamma_j(x)$ escapes to infinity, and $P_j \cap L_i$ is infinite for each $i, j$. Thus the $P_j$ are half-lines that bounce along the blue lines $L_i$ hitting each infinitely many times as they travel 'down' their length off to infinity.

The second image shows that in our example, by excising some initial sub-arcs of the $P_j$'s, and excising some terminal sub-arcs of the $L_i$'s, we can obtain $k$ pairwise-disjoint half-lines each of which 'begins' with an arc from some $L_i$. The question is: For a given $L_k$, is there an $n(k)$ so large that if $P_j$, $1 \leq j \leq n(k)$ are as prescribed then we may pick some $k$ of them and, using the same excision procedure, obtain $k$ pairwise-disjoint half-lines each beginning with an arc from $L$? I have been having some difficulty distilling the question into purely combinatorial terms.

This problem is coming from a problem in topology which I asked on MSE, and as exhibited there this result would be sufficient to prove a nice theorem in continuum theory:

https://math.stackexchange.com/questions/2344525/hairy-points-in-infinite-graphs-and-peano-continua

Thanks for any help! I am useless at these sorts of things.

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