C, then (1, 2) is satisfied by (1, 1) and (0, 3) at Rule 2a (since (0, 3) has at most four neighbors in U , none of them in U 1 ) If (0, 2) ? C, then (1, 2) is satisfied by (1, 1) and (0, 2) at Rule 2a (since C[(0, 1)] = C[(1, 1)] implies that either (?1, 1) ? C or (?1, 2) ? C and consequently (0, 2) has at most four neighbors in U , none of them in U 1 ) and (2, 3) receives charge 1/2 from (1, 3) at Rule 2c, both cases) is satisfied at Rule 2b by (1, 1) and either (0, 3) or (0, 2), and (2, 3) receives charge 1/2 from then u ?2 (v) ? 2, and v satisfies Eq ,

We want to prove by the discharging method that the density of C in V (T k ) is at least 1/4 + 1/4k Every vertex of C begins with charge 1 and every vertex of U begins with charge 0 We say that a vertex (x, y) ? V (T k ) is in the border if y ? {1, k}; otherwise it is in the center If v is in the center, exc(v) = chrg(v) ? 1/4. If v is in the border, exc(v) = chrg(v) ? 3/8. We say that a vertex v is satisfied if exc(v) ? 0. We will prove that, after the application of some discharging rules, every vertex of T k will be satisfied, and we are done, since ((k ? 2) Given a vertex v of T k , let B 1 (v) be the set of neighbors of v in the border and let B 2 (v) be the set of vertices in the border that are at distance 2 of v. When we say that v sends full-excess (resp. half-excess) to B i (v), this means that v sends exc(v)/b (resp. exc(v)/(2b)) to every unsatisfied vertex of B i (v), where b is the number of unsatisfied vertices of B i (v) ,

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