Proof Fix l. WLOG the vertex set of G, say X, is a subset of $\omega $ . We will construct our toast recursively in $\omega $ -stages. At each stage, we will have only finitely many pieces in our toast.

At stage n, if $n \not \in X$ , we do nothing. If $n \in X$ , we check our finitely many pieces to see if any contains $B(n,1)$ . If so, we do nothing.

If not, we find the least $r> 0$ such that $B(n,r)$ can be added to our toast without violating (1). This exists since our toast so far is finite (so, in fact, there is a tail of r’s which will work) and it can be determined effectively since G is highly computable, which means we can computably output the finite graph $G \upharpoonright B(n,r)$ given n and r. We then add $B(n,r)$ to our toast.

Call the resulting set of finite connected sets of vertices $\mathcal {C}$ . $\mathcal {C}$ is computable since at stage n, if we add a new set to $\mathcal {C}$ , that set contains the vertex n. Thus, to test whether a given finite set C is in $\mathcal {C}$ , we only need to run our algorithm until stage $\max (C)$ . (1) is satisfied since we made sure it was satisfied when adding each new piece. For each $n \in X$ , at stage n we made sure there was a piece of toast containing $B(n,1)$ . This implies (2) by induction on the path distance between x and y: If z is a neighbor of y and we have a piece of toast C containing x and z and a piece of toast D containing $B(y,1)$ , then $C \cap D \neq \emptyset $ , so one must contain the other, and then the larger one contains x and y.

Proof If $\Pi $ is not full, then for every nonempty $\mathcal {V}' \subset \mathcal {V}$ and $l' \geq 1$ , there are $l \geq l'$ , multisets, say $a,b \in \mathcal {V}'$ , and labels, say $\alpha , \beta \in \Sigma $ appearing in a and b respectively such that the following holds: Let P be a $\Delta $ -regular path of length l, say with vertices $x_0,\ldots ,x_l$ in order. Label the half edges incident to $x_0$ such that $c(x_0) = a$ and $c(x_0,x_1) = \alpha $ , and likewise for $x_l, b, x_{l-1}$ and $\beta $ respectively. Then if this labeling is extended to a $\Pi $ -coloring c of P, some vertex $x_i$ has $c(x_i) \not \in \mathcal {V}'$ . Note that if $\mathcal {V}' = \mathcal {V}$ , this means there is no such extension.

Since $\Sigma $ is finite, we can say instead that for all nonempty $\mathcal {V}' \subset \mathcal {V}$ , there exists $a,b \in \mathcal {V}'$ and $\alpha ,\beta \in \Sigma $ which have this property for infinitely many l. Fix choices of such $a,b,\alpha $ , and $\beta $ for each $\mathcal {V}'$ , and call them bad for $\mathcal {V}'$ . Also call the infinite set of l witnessing this badness the set of bad l for $\mathcal {V}'$ . Note that determining whether a given l is bad for a given $\mathcal {V}'$ is computable: It can be done by checking all possible colorings of a $\Delta $ -regular path of length l with labels from $\Sigma $ , of which there are only finitely many.

Fix $X,E \subset \text {HF}$ disjoint infinite computable sets, and a computable well order of $\text {HF}$ of type $\omega $ . X and E will be our vertex and edge sets respectively. Fix an effective enumeration $\{\phi _t \mid t \in \omega \}$ of the set of partial recursive functions $X \times E \rightarrow \Sigma $ . (These are candidates for computable colorings of the half edges of our graph.) Fix also an effective enumeration $\{t_n \mid n \in \omega \}$ of $\omega $ which lists each natural number infinitely many times. We are about to describe a recursive construction of $\Delta $ -regular forests $\emptyset = T_0 \subset T_1 \subset \cdots $ with vertex sets contained in X. The vertex set for each $T_n$ will be finite. It will happen that in our construction every $x \in X$ will eventually become a vertex and be made to have degree $\Delta $ . The limit $T = \bigcup _n T_n$ will then be a computable truly $\Delta $ -regular forest. Our construction will be designed so that T does not admit a computable $\Pi $ -coloring.

Alongside T, we will recursively build a T-invariant total computable function $C:X \rightarrow \omega $ . The idea is that the vertices in $C^{-1}(t)$ will be responsible for ensuring that $\phi _t$ is not a $\Pi $ -coloring of T. At stage n, the vertices on which C has been defined will be exactly those in the vertex set of $T_n$ .

At a typical stage in the construction, we will take currently unused (not in the vertex set of our current finite tree) vertices, define C on them, and give them $\Delta $ incident half edges. We will refer to these as “new vertices,” and always choose the least (according to our previously fixed order of $\text {HF}$ ) available. For example, if x has degree 0 in $T_n$ , and in the definition of $T_{n+1}$ we say “add $\Delta $ new vertices as neighbors to x,” it means: let $y_0,\ldots ,y_{\Delta -1}$ be the least elements of X not in the vertex set of $T_n$ , let $e_0,\ldots ,e_{\Delta -1}$ be the virtual edges incident to x in $T_n$ , and add the half edges $(y_i,e_i)$ to $T_{n+1}$ for $i < \Delta -1$ . The exact same treatment will be given to edges. For example, in the previous example, we would probably want to finish by saying “add $\Delta -1$ new virtual edges incident to each $y_i$ ” to maintain $\Delta $ -regularity. This would mean: let $f_{i,j}$ for $i < \Delta $ , $j < \Delta - 1$ be the least elements of E not in the edge set of $T_n$ , and add the half edges $(y_i,f_{i,j})$ to $T_{n+1}$ for each $i,j$ .

In both steps of the previous example, our language leaves some ambiguity about exactly which new vertices/edges are connected to which old edges/vertices. The exact decision here will never matter (except of course in that it should be made in a computable way).

If the stage of our construction is n, C will always be defined to be $t_n$ on new vertices. Thus, our construction will essentially build $\omega $ forests in parallel, one for each t. Let us fix t, and describe the $\omega $ steps in the construction of the t-th forest, i.e., $T \upharpoonright C^{-1}(t)$ .

Fix a very large $N_0\gg |\Sigma |$ to be determined later, independent of t. In the 0-th stage of the construction, place $N_0$ new vertices, say $x_0,\ldots ,x_{N_0-1}$ , in $C^{-1}(t)$ . Also give each $x_i \ \Delta $ new virtual incident edges. This is shown in Figure 1.

Figure 1

Stage 0 of the construction for a fixed t for $\Delta = 3$ .

We also initialize several variables: Set $\mathcal {V}' = \emptyset $ and $N = N_0$ . Also, for each $i < N$ , let $P_i$ be the $\Delta $ -regular path of length 0 whose unique vertex is $x_i$ .

At the start of an arbitrary successor stage, say $m+1$ , suppose we have some $\mathcal {V}' \subset \Sigma $ and $N \in \omega $ very large. Suppose also that we have $\Delta $ -regular paths $P_i$ for $i < N$ and vertices $y_i^c$ for $i < N$ and $c \in \mathcal {V}'$ such that:

1. (1) The $P_i$ ’s and $y_i^c$ ’s all lie in distinct $C^{-1}(t)$ -components.

2. (2) For each i, if $\phi _t$ restricted to the half edges meeting $P_i$ is a $\Pi $ -coloring of $P_i$ , there must be some vertex x of $P_i$ with $\phi _t(x) \not \in \mathcal {V}'$ .

3. (3) $\phi _t(y_i^c) = c$ for each i and c.

Note that this is all satisfied after stage 0.

We start our successor stage by running the Turing machine computing $\phi _t$ for m steps on all of the half edges meeting all of the $P_i$ ’s.

If $\phi _t$ fails to converge in $\leq m$ steps on one of these inputs, or if it converges on all of them but the result fails to be a $\Pi $ -coloring of some $P_i$ , we do what is shown in Figure 2: For each virtual edge e with unique endpoint in $C^{-1}(t)$ , we make e true by adding a new vertex, say x, as its other endpoint, then to maintain $\Delta $ -regularity we add $\Delta -1$ new edges as virtual edges incident to this x. We call this the “uninteresting case.”

Figure 2

The “uninteresting case” for a step of the construction for a fixed t, for $\Delta = 3$ .

The “interesting case” is of course if $\phi _t$ does converge on all these inputs in $\leq m$ steps, and the result is a $\Pi $ -coloring for each $P_i$ . In this case, by condition (2) above, there is a vertex in each $P_i$ , say $z_i$ , with $\phi _t(z_i) \not \in \mathcal {V}'$ . Let $d \in \mathcal {V} \setminus \mathcal {V}'$ such that $\phi _t(z_i) = d$ for at least $N/|\mathcal {V}| \ i$ ’s. Update $\mathcal {V}'$ to $\mathcal {V}' \cup \{d\}$ .

Now $\mathcal {V}'$ is nonempty, so let $a,b \in \mathcal {V}'$ with $\alpha \in a$ and $\beta \in b$ be bad for $\mathcal {V}'$ . Between the $z_i$ ’s and the $y_i^c$ ’s, and using condition (1) above, we can choose degree $\Delta $ vertices $v_i$ and $w_i$ for $i < M := N/(2|\mathcal {V}|)$ such that $\phi _t(v_i) = a$ for each i, $\phi _t(w_i) = b$ for each i, and all $2M$ of these vertices lie in distinct $C^{-1}(t)$ -components. (The factor of $1/2$ is needed for the case $a = b$ .)

Now for each $i < M / 2$ , pick $e_i$ and $f_i$ virtual edges meeting the components of $v_i$ and $w_i$ respectively such that if e is the first edge in the path from $v_i$ to $e_i$ , then $\phi _t(v_i,e) = \alpha $ , and likewise for $w_i$ , $f_i$ , and $\beta $ . Note that these exist since our components are finite $\Delta $ -regular trees. As shown in in the first step in Figure 3, join $e_i$ and $f_i$ with a path of new vertices so that the resulting path between $v_i$ and $w_i$ has a total length which is bad for $\mathcal {V}'$ . (Recall that this is possible as there are arbitrarily long bad lengths, and that determining if a length is bad is computable.) Call this new path $P_i$ . Also, as shown in the second step in Figure 3, add $\Delta -2$ new virtual incident edges to the newly added vertices in $P_i$ to maintain $\Delta $ -regularity. Condition (2) is satisfied for each $P_i$ by definition of bad.

Figure 3

The “interesting case” for a step of the construction for a fixed t. The top and bottom of the image represent the before and after states respectively. The dashed line encloses one of the new paths, $P_i$ .

We can now finish this stage of the construction with the updated value of N being $M/2$ . We still need to define the new $y_i^c$ ’s. Use the unused $v_i$ ’s and $w_i$ ’s for $c \in \{a,b\}$ . For $c \not \in \{a,b\}$ , use the first $M/2 \ z_i$ ’s with $\phi _t(z_i) = d$ if $c = d$ , and the first $M/2 \ y_i^{c}$ ’s from the previous stage of the construction if $c \neq d$ (i.e., if c was in the previous $\mathcal {V}'$ .) (1) is clear by construction.

Thus the description of the construction is complete. We can also see now that the correct value of $N_0$ to take was $N_0> (4|\mathcal {V}|)^{|\mathcal {V}|}$ , since the value of N was divided by $4|\mathcal {V}|$ each time the interesting case occurred, and it cannot occur more than $|\mathcal {V}|$ times for a fixed t (Since $|\mathcal {V}'|$ increases by one each time it does).

Thus T is as desired.

Proof Let G be such a graph, say with vertex and edge sets X and E respectively. WLOG $X \subset \omega $ . Let $\Sigma ' \subset \Sigma $ be greedy. We will recursively define our coloring c in $\omega $ stages. At stage $n \in \omega $ , we will color exactly the half edges incident to n if $n \in X$ , and do nothing otherwise. We will maintain that if $e \in E$ is such that exactly one of the half edges comprising e has been colored so far, then it gets a color from $\Sigma '$ .

We now describe stage n if $n \in X$ : Let $y_i$ for $i < k$ be the neighbors of n less than n. Then by inductive hypothesis, $c(y_i,n) \in \Sigma '$ for each i, and the $\Delta -k$ other incident edges to x, say $e_j$ for $j < \Delta - k$ , not having a $y_i$ as their other endpoint have not been touched. Thus, by the definition of greedy, we may color the half edges incident to x so that $c(x,e_j) \in \Sigma '$ for each j, thus maintaining our inductive hypothesis.

Proof Let G be such a graph, say with vertex set X. By [Reference Bernshteyn2], there is a continuous proper coloring $d:X \rightarrow \omega $ . We can then repeat the construction of the previous proposition, except that at stage $n \in \omega $ we color the half edges incident to all vertices in $d^{-1}(n)$ . This works because each $d^{-1}(n)$ is independent, and the result is continuous since d and G are. (See, for example, Theorem 3.2 in [Reference Brandt, Chang, Grebík, Grunau, Rozhoň and Vidnyánszky5].)

Proof Our construction will closely follow that in the proof of Proposition 2.5. Fix $X,E$ , and the $\phi _t$ ’s as before. Once again, we will be recursively constructing a sequence of $\Delta $ -regular forests with finite vertex sets, but really building a sequence of $\omega $ forests in parallel as kept track of by a recursively defined function $C:X \rightarrow \omega $ . We will use the term “new” for added vertices and edges in the same way. Call the forest we are building T.

Since $\Pi $ is not greedy, for every $\Sigma ' \subset \Sigma $ , there are a $k \in \{0,\ldots ,\Delta \}$ and a sequence $\alpha _0,\ldots ,\alpha _{k-1}$ from $\Sigma '$ witnessing that $\Sigma '$ is not greedy. That is, such that in the situation of Definition 3.1, if this k is used and each half edge $(y_i,x)$ is given the color $\alpha _i$ , then if this is extended to a $\Pi $ -coloring of H, there is some virtual half edge $(x,e)$ such that $c(x,e) \not \in \Sigma '$ . Fix choices of such k and $(\alpha _i)_{i < k}$ for each $\Sigma '$ and call them bad for $\Sigma '$ .

Again, let us fix $t \in \omega $ and describe the $\omega $ steps in the construction of $T \upharpoonright C^{-1}(t)$ .

The 0-th stage will be similar: Fix a very large $N_0\gg |\Sigma |$ to be determined later, independent of t. In the 0-th stage, place $N_0$ new vertices, say $w_0,\ldots ,w_{N_0-1}$ , in $C^{-1}(t)$ , and give each $\Delta $ new virtual incident edges.

We also initialize several variables: Set $\Sigma ' = \emptyset $ and $N = N_0$ . Also set $x_i = w_i$ for each $i < N$ .

At the start of an arbitrary successor stage, say $m+1$ , suppose we have some $\Sigma ' \subset \Sigma $ , $N \in \omega $ very large, and vertices $x_i$ for $i < N$ and virtual half edges $(y_i^{\alpha },e_i^{\alpha })$ for $i < N$ and $\alpha \in \Sigma '$ such that:

1. (1) The $x_i$ ’s and $y_i^{\alpha }$ ’s all lie in distinct $C^{-1}(t)$ -components.

2. (2) For each i, if $\phi _t$ restricted to half edges meeting $B(x_i,1)$ gives a $\Pi $ -coloring, there must be some virtual edge e incident to x with $\phi _t(x,e) \not \in \Sigma '$ .

3. (3) For each i and $\alpha $ , $\phi _t(y_i^{\alpha },e_i^{\alpha }) = \alpha $ .

Note that this is all satisfied after stage 0.

We start our successor stage by running the Turing machine computing $\phi _t$ for m steps on all the half edges meeting each $B(x_i,1)$ . (Note that we can compute $B(x_i,1)$ since our vertex set so far is finite.)

If $\phi _t$ fails to converge in $\leq m$ steps on one of these inputs, or if it converges on all of them but the result fails to be a $\Pi $ -coloring of some $B(x_i,1)$ , we do nothing. We call this the “uninteresting case.”

The “interesting case” is of course if $\phi _t$ does converge on all these inputs in $\leq m$ steps, and the result is a $\Pi $ -coloring for each $B(x_i,1)$ . In this case by condition (2) above, there is a virtual edge $f_i$ for each $i < N$ meeting $x_i$ with $\phi _t(x_i,f_i) \not \in \Sigma '$ . Let $\beta \in \Sigma \setminus \Sigma '$ such that $\phi _t(x_i,f_i) = \beta $ for at least $N/|\Sigma | \ i$ ’s. Update $\Sigma '$ to $\Sigma ' \cup \{\beta \}$ .

Let k and $\alpha _0,\ldots ,\alpha _{k-1} \in \Sigma '$ be bad for $\Sigma '$ . Between the $(x_i,f_i)$ ’s and the $(y_i^{\alpha },e_i^{\alpha })$ ’s, and using condition (1) above, we can chose virtual half edges $(z_i^j,g_i^j)$ for $i < M := N / (\Delta |\Sigma |)$ such that the $z_i^j$ ’s all lie in distinct $C^{-1}(t)$ -components and $\phi _t(z_i^j,g_i^j) = \alpha _j$ for each $j < k$ .

Now for each $i < M/2$ , let $v_i$ be a new vertex and add it as an endpoint to $g_i^j$ for each j, making all these edges true. Then, to maintain $\Delta $ -regularity, add $\Delta -k$ new virtual incident edges to $v_i$ . This is shown in Figure 5.

Figure 5

The “interesting case” for a step of the construction for a fixed t with $\Delta = 3$ . The top and bottom images represent the before and after states respectively.

We can now finish this stage of the construction with the updated value of N being $M/2$ . First set $x_i = v_i$ for each i. Condition(2) is then satisfied by definition of bad. We now define the $(y_i^{\alpha },e_i^{\alpha })$ ’s. If $\alpha = \alpha _j$ for some j, we can use the unused $(z_i^j,g_i^j)$ ’s. Else, if $\alpha = \beta $ we can use the first $M/2 \ (x_i,f_i)$ ’s, and if not, the first $M/2 \ (y_i^{\alpha },e_i^{\alpha })$ ’s from the previous stage of the construction.

Thus the description of the construction is complete. As in the proof of Proposition 2.5, it will be significant that the interesting case cannot occur more than $|\Sigma |$ times for a fixed t (since $|\Sigma '|$ increases by one each time it does). As in that proof, the first application of this is that it tells us what to pick for $N_0$ : We can take $N_0> (2\Delta |\Sigma |)^{|\Sigma |}$ , since $2\Delta |\Sigma |$ is by how much the value of N was divided each time the interesting case occurred.

Proof If $\Sigma '$ is a size $\Delta +1$ set of vertices on which H induces a clique, then this set clearly witnesses that $\Pi _H$ is greedy.

On the other hand, suppose $\Sigma '$ is a set of vertices witnessing that $\Pi _H$ is greedy. We will produce inductively a sequence $v_0,\ldots ,v_{\Delta }$ on which H induces a clique, maintaining inductively that $v_i \in \Sigma '$ for $i < \Delta $ .

We use the notation of Definition 3.1. Let $k \leq \Delta $ and suppose we have $v_i \in \Sigma '$ for $i < k$ , all pairwise adjacent. Apply the definition with this value of k and $c(y_i,x) = v_i$ for all i. We get an extension to the half edges incident to x in which, by definition of $\Pi _H$ , all these half edges must get the same label, call it $v_k$ , and $v_k$ must be adjacent to each previous $v_i$ . Furthermore, if $k < \Delta $ , x has some incident virtual edge, and so $v_k$ must be in $\Sigma '$ , maintaining our inductive hypothesis.

Claim 3.10. For each $\Delta $ , $H_\Delta $ does not contain a $(\Delta +1)$ -clique.

Proof We proceed by induction on $\Delta $ . $H_2$ , a 5 cycle, does not contain a triangle. Suppose we know that $H_\Delta $ contains no $(\Delta +1)$ -clique, and suppose to the contrary that $H_{\Delta +1}$ contains a $(\Delta +2)$ -clique K. Then since $H_{\Delta +1}$ has only one new vertex compared to $H_\Delta $ , the restriction of K to $H_\Delta $ must give either a $(\Delta +1)$ or $(\Delta +2)$ clique depending on whether $w_{\Delta +1} \in K$ . In either case this contradicts the inductive hypothesis.

Claim 3.11. For each $\Delta $ , $H_\Delta \in \mathrm {CONTINUOUS}$ (for $\Delta $ -regular forests).

Proof For this proof we abandon talk of half edges and switch to the usual notion of a graph. We prove by induction on $\Delta $ that every continuous forest with maximum degree $\leq \Delta $ admits a continuous homomorphism to $H_\Delta $ .

For the base case, let G be a continuous forest on a Polish space X with maximum degree 2. By [Reference Bernshteyn2], we can find a clopen maximal 4-discrete set A. Let $G'$ be the continuous graph $G \upharpoonright (X \setminus A)$ . Also by [Reference Bernshteyn2], let $d:(X \setminus A) \rightarrow \omega $ be a continuous coloring such that each $d^{-1}(n)$ is 7-discrete. Note that each $G'$ -component is a path of length at most 7, and so d labels the vertices of each component with unique labels.

We now define our continuous homomorphism, call it c. Set $c(x) = v_0$ for $x \in A$ . Now consider a $G'$ -component, call it P. P is a path of length at most 7. The endpoints are possibly G-adjacent to points in A, and if both are so, the length of P is at least 3. Observe then that we can always extend c to P while keeping it a homomorphism to $H_2$ : This is trivial if we are not in the case where both endpoints of P are adjacent to a point in A, and if we are in that case, we can alternate between $v_0$ and $v_1$ along P if the length of P is even, or circle the 5-cycle $H_2$ once then do this alternation if the length is odd. (Note that P is long enough to allow this.) Furthermore, since $d \upharpoonright P$ is injective, we can do this in a constructive way using d to break symmetry. That is, we can extend c to a homomorphism $G \upharpoonright H_2$ in such a way that the value of c at a point x depends only on the values of d and the indicator function $\chi _A$ of A on $B(x,N)$ for some large constant N. It follows from this and the continuity of d and $\chi _A$ that c is continuous. (See, for example, Theorem 3.2 in [Reference Brandt, Chang, Grebík, Grunau, Rozhoň and Vidnyánszky5].)

The inductive step is now easy. Suppose we have this result for some $\Delta \geq 2$ , and let G be a continuous forest on a Polish space X with maximum degree $\leq \Delta +1$ . We will define a continuous homomorphism c to $H_{\Delta +1}$ . By [Reference Bernshteyn2] again, we may define c be $w_{\Delta +1}$ on a clopen maximal independent set. Let $G' = G \upharpoonright (X \setminus c^{-1}(w_{\Delta +1}))$ . This is a continuous graph with maximum degree $\leq \Delta $ by maximality of $c^{-1}(w_{\Delta +1})$ . Thus by inductive hypothesis there is a continuous homomorphism $c':G' \upharpoonright H_\Delta $ . Since $w_{\Delta +1}$ is adjacent to every vertex in $H_\Delta $ , we can just extend c to all of X by setting $c \upharpoonright (X \setminus A) = c'$ .