By Daniel J. Velleman
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Extra info for American Mathematical Monthly, volume 117, May 2010
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Kharazishvili and T. Sh. Tetunashvili Many examples are known of subsets of the Euclidean plane R2 which have strange and, sometimes, very paradoxical geometrical properties. Usually, such sets in R2 are obtained by using rather delicate set-theoretical techniques. One of the earliest examples of this kind is due to Mazurkiewicz . Namely, in 1914 he proved the following intriguing statement. Theorem 1. There exists a set X ⊂ R2 such that every line in R2 meets X in exactly two points. Any set X ⊂ R2 with the extraordinary property described in Theorem 1 is called a Mazurkiewicz subset of R2 .
From a given array we can obtain others by permuting the rows, permuting the columns, and changing the symbols (as long as we replace a given symbol by another symbol consistently throughout the array). These operations do not change the length of the longest partial transversal or whether there is a transversal or near-transversal. Euler used this observation in his paper of 1779. The following conjecture, which covers all possibilities for the number of rows and the number of columns, is new, though, as we will see, parts of it have already been confirmed.