Mycielskian Trees: Construction and Combinatorial Analysis

• Mycielskian trees are graphs obtained by applying the classical Mycielskian construction to a tree, duplicating its vertices and adding an apex to form 2n+1 vertices.
• The construction preserves original tree edges while introducing new combinatorial features that facilitate analysis of general position numbers and partition enumeration via Bell and Stirling numbers.
• These trees connect advanced set-theoretic methods, including forcing techniques, with practical combinatorial enumeration and structural partition results in graph theory.

A Mycielskian tree is the result of applying the classical Mycielskian construction to a tree, yielding a graph with distinctive combinatorial, structural, and partition properties. These structures underpin generalizations of Mycielski-type theorems in descriptive set theory and connect with fundamental combinatorial enumeration via Bell and Stirling numbers, as well as with extremal and partition properties in graph theory.

1. Definition and Construction

Let $T=(V,E)$ be a tree on $n=|V|$ vertices. The Mycielskian $M(T)$ is constructed as follows:

• Vertex Set: $V \cup V' \cup \{u\}$, with $V'=\{v':v\in V\}$ a disjoint “primed” copy and $u$ a new apex vertex. Thus, $M(T)$ always has $2n+1$ vertices.
• Edge Set: Every edge in $T$ is preserved among the originals. For each edge $\{v,w\}\in E$, one adds the edge $\{v',w\}$. Each primed vertex $v'$ is joined to every original neighbor of $v$ and to the apex $u$. No other edges exist.

This explicit duplication of vertices and edges yields a graph in which the only non-adjacencies are between a vertex $v$ and its copy $v'$, and between non-adjacent pairs in $T$ (which remain so in $M(T)$ unless connected via a shared neighbor).

2. Set-Theoretic Mycielski-Type Theorems Among Trees

Mycielski’s classical theorem asserts that any comeager (or conull) subset $X\subseteq [0,1]^2$ contains a perfect set $P$ with $P\times P\subseteq X\cup\Delta$, where $\Delta$ is the diagonal. Michalski–Rałowski–Żeberski generalized this to rectangles of the form $[A]\times [B]$ in Baire and Cantor spaces, where $A$ and $B$ are “bodies” (infinite branches) of trees of specific types.

Key Definitions:

• Tree body: $$[T]=\{x\in A^\omega:\ \forall n, x\restriction n\in T\}$$, for pruned tree $T\subseteq A^{<\omega}$.
• Miller tree: $T\subseteq\omega^{<\omega}$ such that every node has an extension in some full $\omega$-branching split node.
• Silver tree: $T\subseteq 2^{<\omega}$ perfect; for all pairs $s,t$ at equal length, splitting occurs synchronously.
• Uniformly perfect tree: For each level, either all nodes split or none do.

Main Theorems (Michalski et al., 2019):

• Category (comeager) version: For any comeager $G\subseteq\omega^\omega\times\omega^\omega$, there exist a Miller tree $M$ and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq G\cup\Delta$, with $P$ necessarily not Miller.
• Measure–full version: For any full-measure $F\subseteq 2^\omega\times 2^\omega$, there is a uniformly perfect tree $P$ such that $[P]\times [P]\subseteq F\cup\Delta$, with neither side the body of a Miller or Silver tree.

These results encode sharp constraints: rectangles formed from tree bodies can’t be universally extended to Miller or Silver trees on both sides under measure and category hypotheses.

3. General Position Number and Partition Structure

For a tree $T$, $\mathrm{gp}(M(T))$—the general position number of the Mycielskian—is the cardinality of the largest vertex set in $M(T)$ not containing any three members on a shortest path.

Fundamental bounds (Thomas et al., 2022):

• Lower Bound: $\mathrm{gp}(M(T))\ge\max\{n,2\,\mathrm{ip}_4(T)\}$, where $\mathrm{ip}_4(T)$ is the size of a largest independent set with no three elements meeting a shortest path of length $\le4$.
• Upper Bound: $$\mathrm{gp}(M(T)) \le n+\max\{0,\mathrm{ip}_4(T)-\delta+1\}$$ ($\delta=$ minimum degree), with the universal bound $\mathrm{gp}(M(T))\le n+\alpha(T)-1$ for independence number $\alpha(T)$.

This duality yields the following taxonomy:

• Meagre trees: Trees for which the lower bound is tight.
• Abundant trees: Trees strictly exceeding the lower bound.

Exact formulas were derived for several classes of trees. For instance, if the minimal distance $D_T$ between any two leaves attached to distinct support vertices in $T$ satisfies $D_T\geq5$, then $\mathrm{gp}(M(T))=n+\ell-s$ where $\ell$ is the number of leaves and $s$ is the number of support vertices.

Examples:

Tree class	Order $n$	Leaves $\ell$	Supports $s$	$\mathrm{gp}(M(T))$ formula
Star $S_n=K_{1,n-1}$	$n$	$n-1$	$1$	$2(n-1)=n+\ell-s$
Path $P_n$ ($n\ge4$)	$n$	$2$	$2$	$n$
Caterpillar, full supp	$n$	$\ell$	$s$	$2\ell=n+\ell-s$

These extremal and illustrative cases demonstrate how support-vertex configuration decisively controls general position structure.

4. Bell Numbers and Stirling Numbers: Enumerative Structure

The graphical Bell number $B(M(T))$ counts the number of proper partitions (i.e., into independent sets) of the vertex set of the Mycielskian of $T$, while graphical Stirling numbers $S(M(T);k)$ enumerate those partitions into exactly $k$ blocks.

General formula (Allagan et al., 7 Dec 2025):

$$B(M(T)) = \sum_{A \subseteq V} 2^{|A|} \sum_{B\subseteq A} (-1)^{|B|} B(T[V \setminus (A \cup B)])$$

where $A$ selects original vertices to place with $u$ in a block, and $B$ is the set whose primed counterparts must be excluded by inclusion–exclusion.

Special star graph evaluations:

$B(M(St_n); 3) = 2^n + 1$

$B(M(St_n); 2n) = 2n^2 - 3n + 3$

These correspond precisely to OEIS sequences A000051 and A096376.

Stirling numbers:

$$S(M(T); k) = \sum_{A\subseteq V} 2^{|A|} \sum_{B\subseteq A} (-1)^{|B|} S(T[V\setminus(A\cup B)];\,k-(\,1+|A|\,))$$

This reduces enumeration for any Mycielskian tree to subset sums over induced subtrees, woven together by inclusion–exclusion.

5. Connections to Integer Sequences and Combinatorial Phenomena

Graphical Bell and Stirling numbers for Mycielskian trees exhibit deep links to integer sequences:

Graph/family	Closed Form	OEIS Sequence
$B(M(St_n); 3)$	$2^n + 1$	A000051
$B(M(St_n);2n)$	$2n^2 - 3n + 3$	A096376, A116735
$B(K_{n,n}-M)$	$\sum_{k=0}^n \binom{n}{k} \bell{k}^2$	A385437
$B(K_{n,n};4)$	$2-2^{n+1}+3^{n-1}+4^{n-1}$	A384980
$B(K_{n,n,n};5)$	$(18-18\cdot 2^n+2\cdot 3^n+3\cdot 4^n)/4$	A384988

These identities arise from matching combinatorial binomial–Stirling sums to exponential and polynomial expressions catalogued by the OEIS, confirming the enumerative structure of Mycielskian trees coincides with fundamental patterns in combinatorics and graph enumeration (Allagan et al., 7 Dec 2025).

6. Descriptive Set-Theoretic and Forcing Techniques

The extension from the classical Mycielski theorem to tree rectangles employs advanced set-theoretic tools:

• Category case: Inductive construction of tree families inside countable intersections of dense open sets, guaranteeing the existence of Miller and uniformly perfect subtrees forming the desired rectangle.
• Measure case: Tree-building via density points inside rapidly shrinking clopen sets, ensuring high measure at every stage and the uniform perfectness constraint.
• Forcing/Absoluteness: Use of Cohen-forcing extensions alongside Shoenfield Absoluteness to derive existence of perfect “slalom” trees in any model, underpinning nonstandard proofs and independence arguments (Michalski et al., 2019).

A plausible implication is that the fine partition properties and rectangle-embedding results extend to general families of subtrees, subject to combinatorial constraints, within both category and measure frameworks.

7. Significance, Surprising Behaviours, and Open Directions

Mycielskian trees synthesize set-theoretic, combinatorial, and enumerative phenomena:

• Descriptive set theory: Embedding of highly structured tree rectangles in comeager/conull sets is governed by branching and splitting patterns, with limitations imposing strict constraints on the nature of trees appearing as rectangle sides.
• Combinatorial structure: The behaviour of the general position number under Mycielskian extension varies dramatically, with stars showing a linear increase and paths no change, reflecting the architecture of underlying support-vertex configurations.
• Enumeration: Uniform formulas enable calculation of Bell and Stirling numbers for arbitrary Mycielskian trees. Special cases resolve to canonical integer sequences.
• Surprising patterns: In trees with large diameter, abundance is not guaranteed; intricate dependencies exist on support-vertex and leaf separation. Some caterpillars remain meagre despite the presence of many leaves.

Current results generalize classical Mycielski-type inclusion and partition theorems, revealing new layers of combinatorial and descriptive-set-theoretic interaction. The subject remains open to further structural characterization, deeper probabilistic analysis of tree partitions, and extension to richer families such as Silver and Miller trees under measure and category constraints.