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Ok, I think there are examples where $\Omega(\log n)$ colors are needed.

Here’s an example, let $a_{ij} = \frac{1}{i}$ for $j < i$ and $a_{ij} = \frac{1}{j^2}$ for $j > i$. Then $\sum_{j} a_{ij} = \frac{i-1}{i} + \sum_{j > i} \frac{1}{j^2} = O(1)$. Of course, the bound is $O(1)$ instead of $1$, but that can be normalized and all that.

However, note that $\sum_j a_{1ja_{j1} = \Omega(\log n)$ and if we only have $o(\log n)$ partitions, this sum cannot be "distributed" into small enough parts.

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Ok, I think there are examples where $\Omega(\log n)$ colors are needed.

Here’s an example, let $a_{ij} = \frac{1}{i}$ for $j < i$ and $a_{ij} = \frac{1}{j^2}$ for $j > i$. Then $\sum_{j} a_{ij} = \frac{i-1}{i} + \sum_{j > i} \frac{1}{j^2} = O(1)$. Of course, the bound is $O(1)$ instead of $1$, but that can be normalized and all that.

However, note that $\sum_j a_{1j} = \Omega(\log n)$ and if we only have $o(\log n)$ partitions, this sum cannot be "distributed" into small enough parts.