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.
Motivation.
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. 
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. 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. 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.
Definition.
It is common in better-quasi-ordering theory to write formula_1 for the sequence formula_2 with the first term omitted. Write formula_3 for the set of finite, strictly increasing sequences with terms in formula_4, and define a relation formula_5 on formula_3 as follows: formula_7 if there is formula_8 such that formula_9 is a strict initial segment of formula_10 and formula_11. The relation formula_5 is not transitive.
A "block" formula_13 is an infinite subset of formula_3 that contains an initial segment of every
infinite subset of formula_15. For a quasi-order formula_16, a "formula_16-pattern" is a function from some block formula_13 into formula_16. A formula_16-pattern formula_21 is said to be "bad" if formula_22 for every pair formula_23 such that formula_7; otherwise formula_25 is "good". A quasi-ordering formula_16 is called a "better-quasi-ordering" if there is no bad formula_16-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 formula_28. A "formula_16-array" is a formula_16-pattern whose domain is a barrier. By observing that every block contains a barrier, one sees that formula_16 is a better-quasi-ordering if and only if there is no bad formula_16-array.
Simpson's alternative definition.
Simpson introduced an alternative definition of "better-quasi-ordering" in terms of Borel functions formula_33, where formula_34, the set of infinite subsets of formula_4, is given the usual product topology.
Let "formula_16" be a quasi-ordering and endow formula_16 with the discrete topology. A "formula_16-array" is a Borel function formula_39 for some infinite subset formula_40 of formula_4. A formula_16-array formula_25 is "bad" if formula_44 for every formula_45;
formula_25 is "good" otherwise. The quasi-ordering formula_16 is a "better-quasi-ordering" if there is no bad formula_16-array in this sense.
Major theorems.
Many major results in better-quasi-ordering theory are consequences of the Minimal Bad Array Lemma, which appears in Simpson's paper as follows. See also Laver's paper, where the Minimal Bad Array Lemma was first stated as a result. The technique was present in Nash-Williams' original 1965 paper.
Suppose formula_49 is a quasi-order. A "partial ranking" formula_50 of formula_16 is a well-founded partial ordering of formula_16 such that formula_53. For bad formula_16-arrays (in the sense of Simpson) formula_55 and formula_56, define:
 formula_57
 formula_58
We say a bad formula_16-array formula_60 is "minimal bad" (with respect to the partial ranking formula_50) if there is no bad formula_16-array formula_25 such that formula_64.
The definitions of formula_65 and formula_66 depend on a partial ranking formula_50 of formula_16. The relation formula_69 is not the strict part of the relation formula_65.
Theorem (Minimal Bad Array Lemma). Let formula_16 be a quasi-order equipped with a partial ranking and suppose formula_25 is a bad formula_16-array. Then there is a minimal bad formula_16-array formula_60 such that formula_76.
References.
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