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11 votes

The length of the longest consecutive string of heads or tails that occur asymptotically almost surely when a unbiased coin is flipped repeatedly

The Erdos-Renyi law of large numbers answers this, in a strong sense and even in a more generalized fashion. More recent work in this direction includes papers by Arratia, Gordon, Waterman (see here) ...
kodlu's user avatar
  • 9,538
7 votes

counting fixed-area closed walks on square 2d lattice

This is a difficult question, and the answer depends on what exactly do you want to know. If you want to know asymptotics for $A$ fixed, then the answer is known for $A=0$ and I would guess it's the ...
Igor Pak's user avatar
  • 16.1k
3 votes

Number of solutions of $am \equiv bn \pmod{q}$

I made a fatal mistake in my attempt at obtaining a simple asymptotic formula for the number $J$ of solutions to the congruence with the given restrictions. Undaunted, I come back with the proof of a ...
José Hdz. Stgo.'s user avatar
2 votes

How to maximize the number of people that mingle with each other?

This is the problem of finding optimal partial edge clique coverings. See Damaschke, P. Optimal partial clique edge covering guided by potential energy minimization. Optim Lett 13, 1469–1481 (2019). ...
RobPratt's user avatar
  • 4,236
1 vote

The vertex-covering number of a particular hypergraph

I am afraid that $f(n)$ does not grow, even if you use only permutations of $(0,1,2)$. I claim that the minimal size of $S$ is $3^{n-1}$. As an example, you may take all rows with first coordinate 0. ...
Fedor Petrov's user avatar
  • 94.7k
1 vote

Can one make Erdős's Ramsey lower bound explicit?

I just saw this. Actually there is a constructive solution to the original Ramsey number lower bound. One can define an energy function which counts the number of mono-chromatics and try to minimize ...
CHUAKS's user avatar
  • 731

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