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Noel, Jonathan A.
Weak saturation of the cube in the clique ★
Determine .
Keywords: bootstrap percolation; hypercube; Weak saturation
Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★
![$ 4 $](/files/tex/1f1498726bb4b7754ca36de46c0ccdd09136d115.png)
Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation
Saturation in the Hypercube ★★
Author(s): Morrison; Noel; Scott
![$ 2\ell $](/files/tex/e6160c4357fdf2ec5854a3cc78837f8a67caa5c5.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
Keywords: cycles; hypercube; minimum saturation; saturation
Cycles in Graphs of Large Chromatic Number ★★
Author(s): Brewster; McGuinness; Moore; Noel
![$ \chi(G)>k $](/files/tex/84d787e716a616f1d9b6d33aea0d9f0777cb1df3.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \frac{(k+1)(k-1)!}{2} $](/files/tex/2576c24e6815c0bf97ab23f18ad24cf5421aeac4.png)
![$ 0\bmod k $](/files/tex/ef4d29155ecd56ddbfea81561d000d0e5823edb7.png)
Keywords: chromatic number; cycles
Saturated $k$-Sperner Systems of Minimum Size ★★
Author(s): Morrison; Noel; Scott
![$ 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} $](/files/tex/1564812bc34ed3ebd060debb561be85acfb45f10.png)
Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system
Partitioning the Projective Plane ★★
Author(s): Noel
Throughout this post, by projective plane we mean the set of all lines through the origin in .
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ S'\subseteq S $](/files/tex/e8109a0f5c1dcb11fe3245b53b8bd2bc9d6418d1.png)
![$ |S'|=3 $](/files/tex/e1b8bc4df405ab37c6b4aa2f638a5c2df882f7a9.png)
![$ S_1,S_2,S_3 $](/files/tex/6f2284a502ee75f719fa3d5c2430c467e11df0c4.png)
![$ S_4 $](/files/tex/a0fc8ce0b0dfbf88309c7c045fff90a5cadd5117.png)
![$ S_i $](/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png)
![$ S_i $](/files/tex/110ae457d97eebe47aa4d2e8c6237fdb9317f11e.png)
Keywords: Partitioning; projective plane
Choosability of Graph Powers ★★
Author(s): Noel
![$ f(k)=o(k^2) $](/files/tex/bd642e5dd66f1577cedf5fed57f75187a80168ac.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![\[\text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\right)\right)?\]](/files/tex/989db06683633e86605c26e7d9f0bffc7e46a496.png)
Keywords: choosability; chromatic number; list coloring; square of a graph
Choice Number of k-Chromatic Graphs of Bounded Order ★★
Author(s): Noel
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ mk $](/files/tex/c6307bea3dbafaf140155114bd35bb788fc000d8.png)
![$ \text{ch}(G)\leq \text{ch}(K_{m*k}) $](/files/tex/0d0a9568608331000410c601a041b1c391b31d01.png)
Keywords: choosability; complete multipartite graph; list coloring
Mixing Circular Colourings ★
![$ \mathfrak{M}_c(G) $](/files/tex/f24f4a47b3db2823b31406ff6f37ecb0296f4f10.png)
Keywords: discrete homotopy; graph colourings; mixing
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