Erdős Discrepancy Problem — Can $\pm 1$ Subsequence Sums Stay Bounded?

For any sequence $x_1, x_2, \ldots \in \{+1, -1\}$, do the partial sums $S(n, d) = \sum_{k=1}^n x_{kd}$ always become arbitrarily large? Erdős (1932) conjectured yes — no $\pm 1$ sequence can keep all "arithmetic-progression sums" bounded.

For $d=1$, the sums resemble a Bernoulli walk; all $d$ together impose multiplicative structure. Polymath5 found discrepancy-2 sequences of length 1124 and useful experimental structure. Konev and Lisitsa then certified by SAT in 2014 that length 1160 is possible but 1161 is not. That finite $C=2$ result is distinct from the later theorem for every bound $C$.

Tao completely resolved the problem in 2015 (Discrete Analysis, 2016) by reducing it to a correlation problem for multiplicative functions and using entropy decrement. Polymath5 supplied experiments and partial structure; the exact length-1160/1161 result was a separate SAT proof by Konev and Lisitsa. Neither is the same as the all-$C$ theorem.