Geodesic
On Disjoint Connected Subsets of a Square Containing Pairs of Antipodal Points
Canadian mathematical bulletin, Tome 9 (1966) no. 5, pp. 631-638

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In [1], p. 132, it is shown that there exist two disjoint subsets of a square each connecting a pair of opposite vertices. Our interest in obtaining covers by infinite collections of such sets was prompted by a question posed by R.B. Reed in [3], We are grateful also to Dr. M. Edelstein for his guidance and encouragement of our work on the subject. 
Morrison, D.; Kiang, M.; Wright, J. On Disjoint Connected Subsets of a Square Containing Pairs of Antipodal Points. Canadian mathematical bulletin, Tome 9 (1966) no. 5, pp. 631-638. doi: 10.4153/CMB-1966-076-8
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     author = {Morrison, D. and Kiang, M. and Wright, J.},
     title = {On {Disjoint} {Connected} {Subsets} of a {Square} {Containing} {Pairs} of {Antipodal} {Points}},
     journal = {Canadian mathematical bulletin},
     pages = {631--638},
     year = {1966},
     volume = {9},
     number = {5},
     doi = {10.4153/CMB-1966-076-8},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1966-076-8/}
}
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