Lemma 30 ([Deleting a history of paths]). Let h(G)<t . Start with F_0=W_0 , where |W_0|=t , and perform at most R_t rounds with the following rules. At the start of a round there is a footprint F and a set W⊆F of t tips. Choose disjoint …

We use c_d from Lemma 15 ([Additive list-coloring loss]) and B_* from Lemma 17. All constants and cutoffs in this proof are independent of the history and of the lengths of its paths. The two path collections. Remove the starting vertex from each first-system path, and collect all the resulting nonempty paths in a family A . These paths are pairwise vertex-disjoint. Within one round the only intersections of first-system paths are their prescribed starts, which have been removed. Across rounds, a new first system avoids the old footprint except at its tips W . Every tip is a prescribed start, and no such start is internal on any path of the system. After stripping starts, the new paths therefore avoid the entire old footprint. Let B consist of the full second-system paths from all rounds. These too are pairwise vertex-disjoint: each new second system avoids the entire old footprint, which contains every earlier second-system path. No disjointness between A and B is asserted or needed. There are at most 4t first paths and 2t second paths per round, so, if q≤R_t rounds have been started, | A|+| B|≤6tq≤6tR_t. Small sets and exceptional vertices. Let X_0 contain all districts J,D , all tip sets appearing in the history, and the ends of every path in both systems. Counting repetitions only increases the following bound: |X_0|≤2R_t t^ 1.06 +(13R_t+1)t. Indeed, there are at most R_t+1 tip sets; the two systems together have at most 12t path-end occurrences per round. Consider the footprints at the start of each round and also F_ last ; call these at most R_t+1 sets the snapshots. The final snapshot will control neighbors of vertices outside the final footprint. Earlier snapshots will control the neighbors that a newly entering vertex has on older paths. At a snapshot T , let A_T, B_T be the two collections formed by systems already included in T . They use the original recorded paths, with only each first path's own start removed; no vertices of the set X defined below have yet been removed from these collections. For either collection C , declare a vertex v∈V(G) T exceptional if it has a neighbor on at least B_*t distinct members of C . To count these vertices, contract the disjoint connected paths of C to a set of | C| vertices. Vertices outside T are not contracted. Their numbers of distinct neighbors in the contracted set are exactly the numbers of paths they see. The resulting graph is a minor of G , so Lemma 17 implies that the number of exceptional vertices is at most t^ 1+2ε | C| B_*t ≤t^ 2ε | C|. Apply this separately to A_T and B_T . By, the number at one snapshot is at most 6R_t t^ 1+2ε . Let X be X_0 together with the exceptional vertices from all snapshots, including exceptional vertices outside F_ last . Then |X| , ≤2R_t t^ 1.06 +(13R_t+1)t +6R_t(R_t+1)t^ 1+2ε ; , =O(t^ 1.06 ^4t)+O(t^ 1.01 ^8t) ≤t^ 11/10 for sufficiently large t , since ε=1/200 . Thus every required snapshot has been counted. Put Y=F_ last X . The sets X,Y are disjoint, and every vertex of Y lies on a recorded path. Neighbors from outside the footprint. Let v∉X∪Y . Then v∉F_ last , so it was outside every recorded path union. Moreover, it belonged to the graph in which each system was minimized: every excluded set F W or F∪R is contained in F_ last . By Lemma 27 ([Bounds from unpaired path minimization]), it has at most three neighbors on each path. Nonexceptionality at the final snapshot bounds the number of paths it sees by fewer than B_*t in each collection. All stripped starts lie in X_0 . Consequently |N_G(v)∩Y|≤6B_*t. Degeneracy inside the footprint. Order the vertices of Y by their round of first entry into the footprint. Within a round, put vertices on Q_1 first, and then the remaining new vertices on Q_2 . This partitions Y into at most two phases per round, because district vertices and all initial tips lie in X_0 . A vertex v entering in a given round was outside that round's old footprint F . If v∈Y , it was not exceptional at the snapshot F . For every earlier system, its forbidden vertices belong to F , while v∉F ; hence v was eligible in the graph of that minimization and was outside its path union. The preceding argument therefore gives at most 6B_*t neighbors of v in Y from earlier rounds. A first phase is contained in the union of at most 4t first-system paths with their starts removed. Lemma~ Lemma 27 ([Bounds from unpaired path minimization]) makes its induced graph 8t -degenerate. A second phase is contained in the union of 2t minimizing second-system paths, so its induced graph is 4t -degenerate, and in particular 8t -degenerate. These are assertions about the induced graphs in G : all vertices of the relevant phase belong to the graph used for that minimization, which was an induced subgraph of G . Finally, a vertex in the second phase of a round lies outside the entire first-system union and outside the old footprint. It therefore belongs to G-(F W) . Lemma 27 ([Bounds from unpaired path minimization]) bounds its neighbors in the first phase by 3(4t)=12t . In any nonempty subgraph of G[Y] , select the latest represented phase and a vertex of degree at most 8t within that phase. It has no neighbors in a later represented phase, at most 12t neighbors in an earlier phase of the same round, and at most 6B_*t neighbors in earlier rounds. Hence G[Y] is (6B_*+20)t -degenerate. Equations and now verify every hypothesis of Lemma 15 ([Additive list-coloring loss]), with d=6B_*+20 . It follows that ℓ(G-X-Y)≥ℓ(G)-c_ft. Since F_ last ⊆X∪Y , the graph G-F_ last contains G-X-Y as an induced subgraph. Monotonicity of list chromatic number proves the assertion. For the initial footprint the same proof has Y=∅ and X=W_0 .