Almost nothing is known about the parity of the partition function p(n), which is conjectured to be random. Despite this expectation, Ono [10] surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4 for p(n), indicating that the parity of the partition function cannot be truly random. Answering a question of Ono, we explicitly exhibit the first examples of these relations which he proved theoretically exist. The first two relations invoke 131 (resp. 198) different discriminants \( D \le 24 k - 1 \) for \( k = 309 \) (resp. \( k = 312 \) ); new relations occur for \( k = 316,\, 317,\, 319,\, 321,\, 322,\, 326, \, \ldots \) .
