This is my first ever post so I hope this is an appropriate question.
Basically I am looking at the paper here: http://homepages.math.uic.edu/~mubayi/papers/biclique.pdf Namely theorem 5.
Now, feel free to read it of course but to summerise the salient details to ask my question:
We are colouring a graph $G$. The colour classes are sequences with each entry of the sequence a term in $\{1, \dots, m\}$. Note that different colour classes may have different lengths. Now, the paper doesn't worry about $\log m$ being an integer, but I think in context of what the colouring is doing I think it has to be. Anyway, the paper reasons that "the maximum length colour sequence is length log(m)". I reason "for $m$ of the form $m=2^n -1$, the maximum length colour sequence is length $\lceil \log_{2}m \rceil$. For any other $m$, the maaximum length colour sequence is length $\lfloor \log_{2}m \rfloor$. Now, the paper that concludes that $\chi(G) \le m^{\log_{2}m}(1+o(1))$. It's also worth noting that this is for $\chi(G)$ large and $m$ grows with $\chi(G)$.
Is the paper correct to conclude how it does? Don't we have to say that $\chi(G) \le m^{\lceil \log_{2}m \rceil}(1+o(1))$? Or is this the same thing?
Any help appreciated and again I hope this is the correct place to ask this sort of question.