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Morrison, Natasha
Weak saturation of the cube in the clique ★
Problem
Determine .
Keywords: bootstrap percolation; hypercube; Weak saturation
Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★
Problem Determine the smallest percolating set for the
-neighbour bootstrap process in the hypercube.
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation
Saturation in the Hypercube ★★
Author(s): Morrison; Noel; Scott
Question What is the saturation number of cycles of length
in the
-dimensional hypercube?
![$ 2\ell $](/files/tex/e6160c4357fdf2ec5854a3cc78837f8a67caa5c5.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
Keywords: cycles; hypercube; minimum saturation; saturation
Saturated $k$-Sperner Systems of Minimum Size ★★
Author(s): Morrison; Noel; Scott
Question Does there exist a constant
and a function
such that if
, then every saturated
-Sperner system
has cardinality at least
?
![$ c>1/2 $](/files/tex/71b371b57345af0cbcfd2ffd78362b8988723a7d.png)
![$ n_0(k) $](/files/tex/77717a7ba8441af96e47411a5b9a5d4a913f3dba.png)
![$ |X|\geq n_0(k) $](/files/tex/e6c6b51d4e2df6fd85840bc289e38c981674e057.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ \mathcal{F}\subseteq \mathcal{P}(X) $](/files/tex/cb32025ab3209d1516fd6ea63a4d8eb206a81411.png)
![$ 2^{(1+o(1))ck} $](/files/tex/1564812bc34ed3ebd060debb561be85acfb45f10.png)
Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system
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