bio  website  nd.edu/~lnicolae 

location  University of Notre Dame  
age  49  
visits  member for  2 years, 11 months 
seen  19 mins ago  
stats  profile views  5,842 
I do mostly geometry and topology, with an analytic bias. For the past few years Morse theory has popped up in my research, but not in a conventional way.
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A game of stones
I think that in your claim(s) you need to require that $q+1\geq 3$. 
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accepted  A game of stones 
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@AaronMeyerowitz That is nice! 
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@IanMorris I could not follow the Game of Thrones. After a season there were too many characters to process. However, I book titled A storm of surds sounds very intriguing. 
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awarded  Popular Question 
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@Per Alexandersson I have added a remark to my original question shown that the nature of the jump $n\to n^2$ ought to play a role if the answer is positive. 
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awarded  Nice Question 
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@Per your argument is an induction over the number $k$ of overcrowded nodes. What happens when there is exactly one overcrowded node, with lots of stones. 
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@Andrew This is the key case. Maybe Per ought to add some details. 
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Your argument seems to work with $n^2$ replaced by a function $f(n)>n+1$. 
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@ Anthony Yes, you are right. 
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Thanks Vidit. I will have a look at that post. 
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@ Matthias $s$ and $N$ are independent, and $N$ is meant to be very large. 
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asked  A game of stones 