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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.
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).
A real number x is called an upper bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real ...
An upper bound for R(r, s) can be extracted from the proof of the theorem, and other arguments give lower bounds. (The first exponential lower bound was obtained by Paul Erdős using the probabilistic method.) However, there is a vast gap between the tightest lower bounds and the tightest upper bounds.
Toggle Upper and lower bounds subsection. 2.1 General placement methods. 2.2 Upper bound. 2.3 Conjectured bounds. 3 Applications. ... The numbers of solutions with ...
The lower bound was given by an easy argument. The upper bound is given by a n × n {\displaystyle {\sqrt {n}}\times {\sqrt {n}}} square grid. For such a grid, there are O ( n / log n ) {\displaystyle O(n/{\sqrt {\log n}})} numbers below n which are sums of two squares, expressed in big O notation ; see Landau–Ramanujan constant .
This problem can be found amongst the problems proposed by Paul Erdős in combinatorial number theory, known by English speakers as the Minimum overlap problem.It was first formulated in the 1955 article Some remarks on number theory [3] (in Hebrew) in Riveon Lematematica, and has become one of the classical problems described by Richard K. Guy in his book Unsolved problems in number theory.
Therefore, if one can show a lower bound for (/,;,) that matches the upper bound up to a constant, then by a simple sampling argument (on either an / bipartite graph or an / bipartite graph that achieves the maximum edge number), we can show that for all ,, one of the above two upper bounds is tight up to a constant.