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2. Upper and lower bounds - Wikipedia

   en.wikipedia.org/wiki/Upper_and_lower_bounds

   The set S = {42} has 42 as both an upper bound and a lower bound; all other numbers are either an upper bound or a lower bound for that S. Every subset of the natural numbers has a lower bound since the natural numbers have a least element (0 or 1, depending on convention). An infinite subset of the natural numbers cannot be bounded from above.

3. Coin problem - Wikipedia

   en.wikipedia.org/wiki/Coin_problem

   Simpler lower and upper bounds for Frobenius numbers for n = 3 have also been determined. The asymptotic lower bound due to Davison The asymptotic lower bound due to Davison f ( a 1 , a 2 , a 3 ) ≡ g ( a 1 , a 2 , a 3 ) + a 1 + a 2 + a 3 ≥ 3 a 1 a 2 a 3 {\displaystyle f(a_{1},a_{2},a_{3})\equiv g(a_{1},a_{2},a_{3})+a_{1}+a_{2}+a_{3}\geq ...

4. Square-difference-free set - Wikipedia

   en.wikipedia.org/wiki/Square-difference-free_set

   The best known upper bound on the size of a square-difference-free set of numbers up to is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size . Closing the gap between these upper and lower bounds remains an open problem.

5. Glossary of mathematical symbols - Wikipedia

   en.wikipedia.org/wiki/Glossary_of_mathematical...

   Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sorts of mathematical objects. As the number of these sorts has remarkably increased in modern mathematics, the Greek alphabet and some Hebrew letters are

6. Completeness (order theory) - Wikipedia

   en.wikipedia.org/wiki/Completeness_(order_theory)

   But this is just the least element of the whole poset, if it has one, since the empty subset of a poset P is conventionally considered to be both bounded from above and from below, with every element of P being both an upper and lower bound of the empty subset. Other common names for the least element are bottom and zero (0).

7. Roth's theorem on arithmetic progressions - Wikipedia

   en.wikipedia.org/wiki/Roth's_Theorem_on...

   Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...