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The lower bound was given by an easy argument. The upper bound is given by a square grid. For such a grid, there are (/ ) numbers below n which are sums of two squares, expressed in big O notation; see Landau–Ramanujan constant.
13934 and other numbers x such that x ≥ 13934 would be an upper bound for S. 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 ...
The new solution (¯,) is used to update the lower bound. If the gap between the best upper and lower bound is less than then the procedure terminates and the value of ¯ is determined by solving the primal residual problem fixing ¯. Otherwise, the procedure continues on to the next iteration.
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.
A lower bound from simplexes is +. For >, a lower bound of + is available using a generalization of the Moser spindle: a pair of the objects (each two simplexes glued together on a facet) which are joined on one side by a point and the other side by a line. An exponential lower bound was proved by Frankl and Wilson in 1981.
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.
Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set of diameter one. The diameter of a set by definition is the least upper bound of the distances between all pairs of points in the set. A shape covers a set if it contains a congruent subset.
Hammersley stated an upper bound on the sofa constant of at most . [4] [1] [9] Yoav Kallus and Dan Romik published a new upper bound in 2018, capping the sofa constant at . Their approach involves rotating the corridor (rather than the sofa) through a finite sequence of distinct angles (rather than continuously) and using a computer search to ...