- A. W.~Apter and J. D. Hamkins, “Indestructibility and the level-by-level agreement between strong compactness and supercompactness,” Journal of Symbolic Logic, vol. 67, iss. 2, p. 820–840, 2002.

[Bibtex]`@ARTICLE{ApterHamkins2002:LevelByLevel, AUTHOR = {Arthur W.~Apter and Joel David Hamkins}, TITLE = {Indestructibility and the level-by-level agreement between strong compactness and supercompactness}, JOURNAL = {Journal of Symbolic Logic}, VOLUME = {67}, YEAR = {2002}, NUMBER = {2}, PAGES = {820--840}, ISSN = {0022-4812}, CODEN = {JSYLA6}, MRCLASS = {03E35 (03E55)}, MRNUMBER = {1905168 (2003e:03095)}, MRREVIEWER = {Carlos A.~Di Prisco}, DOI = {10.2178/jsl/1190150111}, URL = {http://wp.me/p5M0LV-2i}, eprint = {math/0102086}, archivePrefix = {arXiv}, primaryClass = {math.LO}, }`

Can a supercompact cardinal $\kappa$ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above $\kappa$, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or less level-by-level agreement, such as requiring it only on measure one sets, then yes, it can.