In order theory a **better-quasi-ordering** or **bqo** is a quasi-ordering that does not admit a certain type of bad array. Every better-quasi-ordering is a well-quasi-ordering.

Though *well-quasi-ordering* is an appealing notion, many important infinitary operations do not preserve well-quasi-orderedness.
An example due to Richard Rado illustrates this.^{[1]}
In a 1965 paper Crispin Nash-Williams formulated the stronger notion of *better-quasi-ordering* in order to prove that the class of trees of height ω is well-quasi-ordered under the *topological minor* relation.^{[2]} Since then, many quasi-orderings have been proven to be well-quasi-orderings by proving them to be better-quasi-orderings. For instance, Richard Laver established Laver's theorem (previously a conjecture of Roland Fraïssé) by proving that the class of scattered linear order types is better-quasi-ordered.^{[3]} More recently, Carlos Martinez-Ranero has proven that, under the proper forcing axiom, the class of Aronszajn lines is better-quasi-ordered under the embeddability relation.^{[4]}

It is common in better-quasi-ordering theory to write for the sequence with the first term omitted. Write for the set of finite, strictly increasing sequences with terms in , and define a relation on as follows: if there is such that is a strict initial segment of and . The relation is not transitive.

A *block* is an infinite subset of that contains an initial segment^{[clarification needed]} of every
infinite subset of . For a quasi-order , a *-pattern* is a function from some block into . A -pattern is said to be *bad* if ^{[clarification needed]} for every pair such that ; otherwise is *good*. A quasi-ordering is called a *better-quasi-ordering* if there is no bad -pattern.

In order to make this definition easier to work with, Nash-Williams defines a *barrier* to be a block whose elements are pairwise incomparable under the inclusion relation . A *-array* is a -pattern whose domain is a barrier. By observing that every block contains a barrier, one sees that is a better-quasi-ordering if and only if there is no bad -array.

Simpson introduced an alternative definition of *better-quasi-ordering* in terms of Borel functions , where , the set of infinite subsets of , is given the usual product topology.^{[5]}

Let * be a quasi-ordering and endow with the discrete topology. A **-array* is a Borel function for some infinite subset of . A -array is *bad* if for every ;
is *good* otherwise. The quasi-ordering is a *better-quasi-ordering* if there is no bad -array in this sense.

Many major results in better-quasi-ordering theory are consequences of the Minimal Bad Array Lemma, which appears in Simpson's paper^{[5]} as follows. See also Laver's paper,^{[6]} where the Minimal Bad Array Lemma was first stated as a result. The technique was present in Nash-Williams' original 1965 paper.

Suppose is a quasi-order.^{[clarification needed]} A *partial ranking* of is a well-founded partial ordering of such that . For bad -arrays (in the sense of Simpson) and , define:

We say a bad -array is *minimal bad* (with respect to the partial ranking ) if there is no bad -array such that .
The definitions of and depend on a partial ranking of . The relation is not the strict part of the relation .

**Theorem** (Minimal Bad Array Lemma)**.** Let be a quasi-order equipped with a partial ranking and suppose is a bad -array. Then there is a minimal bad -array such that .