Every finite graph has a bounded partition into self-regular vertex sets

Apply [L2] with old parameter $ϵ\mathrm{/}2$ and new parameter $ϵ$, and let $\tau >0$ be a permitted relative vertex-change fraction. Let $\delta >0$ be supplied by [L1] at parameter $ϵ\mathrm{/}2$.

Repeatedly apply [L1] to the graph induced by the vertices not yet chosen, stopping if that remainder is empty and otherwise removing an $(ϵ\mathrm{/}2)$-self-regular set of at least a $\delta$ fraction of it. Choose a fixed $r$ with $(1-\delta {)}^r\le \tau \mathrm{/}(1+\tau )$. After at most $r$ rounds, the unassigned remainder has size at most $\tau$ times the size of the union of the extracted sets.

Distribute every remaining vertex among the at most $r$ extracted sets proportionally to their sizes. For all sufficiently large graph orders, rounding can be absorbed by replacing $\tau$ in step 1.1 with a smaller constant, so no set grows by more than the vertex-change fraction allowed by [L2]. No old incidence among vertices of an extracted set is changed.

By [L2], every enlarged set is $ϵ$-self-regular. Empty and bounded exceptional graphs may instead be partitioned into singletons, increasing the uniform bound only by a constant; hence some $K(ϵ)$ works for all finite graphs.