A 3-SAT instance asks whether there is a true or false assignment that satisfies all clauses, each containing at most three literals. The hypothesis concerns the growth of worst-case running time as the number of variables increases.
Researchers use reductions to relate other problems to satisfiability. Under the hypothesis, a sufficiently fast algorithm for a related problem would also give a faster algorithm for 3-SAT if the reduction preserves the relevant size bound.
ELI5
This hypothesis says that some growing logic puzzles cannot always be solved without a rapidly growing amount of work. Easy examples do not settle the hardest cases.
For example, a solver may finish many small puzzles quickly while still needing far more work on specially difficult larger ones.
Does this say every puzzle is hard?
No. It is a claim about worst-case running time, and many particular instances can be solved quickly.
Would a refutation settle P versus NP?
A subexponential algorithm can still take more than polynomial time, so such a result alone would not settle P versus NP.
