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Results tagged with extremal-combinatorics
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user 43954
10
votes
Dividing a cake between $n-1$, $n$, or $n+1$ guests
Proposition. $f(n)\ge 2n+\tfrac 13\left(\sqrt{\tfrac{n}3+1}-2\right) $ for any natural $n\ge 13$.
Proof. Fix the cake cutting with the minimum number $f=f(n)$ of slices. We shall work with the graph $ …
- 5,409
4
votes
1
answer
361
views
Orchard-planting problem in space
The original orchard-planting problem asks for the maximum number of $3$-point lines attainable by a configuration of points in the plane. I am interested in its natural generalization for (three-dime …
- 5,409
2
votes
Lower bound for the size of a family of sets
Let $n=|\mathcal F'|$. Below we easily show that $n\ge\sqrt{m}$ and I shall look for more refined arguments to improve this bound.
For each natural $N$ put $[N]=\{1,2,\dots,N\}$. For each $i\in [2]$ p …
- 5,409
1
vote
Bounds for ground set of Steiner system (inverse EKR style problem)
I am still studying relevant sources, trying to provide the required bounds.
So I am going to update this answer accordingly.
If $l>k$ or $r=1$ then the required condition always holds, so we assume t …
- 5,409