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Part of 2015 Ukraine Team Selection Test

Problems(1)

n teams in a football tournament

Source: Ukraine TST 2015 p2

5/2/2020
In a football tournament, nn teams play one round (n⋮2n \vdots 2). In each round should play n/2n / 2 pairs of teams that have not yet played. Schedule of each round takes place before its holding. For which smallest natural kk such that the following situation is possible: after kk tours, making a schedule of k+1k + 1 rounds already is not possible, i.e. these nn teams cannot be divided into n/2n / 2 pairs, in each of which there are teams that have not played in the previous kk rounds.
PS. The 3 vertical dots notation in the first row, I do not know what it means.
combinatorics