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) ...
- 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 ...
- 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 ...
- 8,472
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).
...
- 4,236
1
vote
Accepted
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.
...
- 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 ...
- 731
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