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SQLite 3.7.5 and above versions available in Redhat 6
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Red Hat generally does not update versions of a given package once it's released in the first stable version. All changes are made by backporting fixes from later versions into the same version originally released. This is how they get their uncommon stability: by not replacing an existing version of packaged software with a new one. The versions that were in place upon its initial release on  November 10, 2010 are the versions shipped in the last release of RHEL 6.

(There are sometimes exceptions to this, but they're quite rare, and generally peripheral besides. e.g. Firefox, upon which no core OS software depends.)

However, it is also the case that RHEL 6 has been [completely outside of any support window for nearly half a year](https://en.wikipedia.org/wiki/Red_Hat_Enterprise_Linux#RHEL_6), so even if none of the above were true, there would be no way to get Red Hat to give you an upgrade now. They're just going to tell you to upgrade to RHEL 7 or RHEL 8.

If you insist on remaining on this obsolete platform and also wish to have updated software atop it, you'll have to arrange to build it yourself. There are many paths forward, but you can't reasonably expect someone else to take on this pain of 10 years of technical debt for you.