To solve this problem, we need to maximize the size of the set $T \subset \{1,2,\dots,n\}^3$ under the given constraints. Let's break down the problem step by step.

1. Understanding the Constraints:
- Each triplet $(a, b, c) \in T$ must satisfy $a < b < c$.
- For any two distinct triplets $(a, b, c)$ and $(x, y, z)$ in $T$, at most one of the equalities $a = x$, $b = y$, or $c = z$ can hold.

2. Analyzing the Triplet Structure:
- Since $a < b < c$, each triplet is strictly increasing.
- We need to ensure that no two triplets share more than one coordinate.

3. **Constructing the Set $T$:**
- Consider the set of all possible triplets $(a, b, c)$ where $1 \leq a < b < c \leq n$.
- The number of such triplets is given by the binomial coefficient $\binom{n}{3}$, which counts the number of ways to choose 3 distinct elements from $n$ elements in increasing order.

4. Ensuring the Constraint on Equalities:
- We need to ensure that no two triplets share more than one coordinate.
- One way to achieve this is to select triplets such that no two triplets have the same first, second, or third element.

5. **Maximizing the Size of $T$:**
- To maximize $|T|$, we need to find the largest possible subset of $\binom{n}{3}$ triplets that satisfies the constraints.
- A useful approach is to consider the combinatorial structure and use the pigeonhole principle to ensure that no two triplets share more than one coordinate.

6. **Constructing a Valid Set $T$:**
- One possible construction is to choose triplets such that each element $a, b, c$ appears in a unique position across different triplets.
- For example, if $n = 4$, we can choose the triplets $(1, 2, 3)$, $(1, 3, 4)$, $(2, 3, 4)$, ensuring that no two triplets share more than one coordinate.

7. Generalizing the Construction:
- For a general $n$, we can construct a set $T$ by carefully selecting triplets to avoid sharing more than one coordinate.
- The maximum size of such a set can be shown to be $\boxed{\binom{n}{3}}$, as this is the largest number of triplets that can be chosen without violating the constraints.