More than thirty years ago, Paul Erd?s asked the following question: Does there exist an absolute constant c such that any measur- able planar region R of area c contains the vertices of a unit-area triangle ? In traditional Erd?s style, the question stood the test of time and despite the efforts of many mathematicians it is still wide open. We give a positive answer with c = 41n2 in two particu- lar cases: ? Case 1: there exists a direction l such that the projection of R onto l is an interval. ? Case 2: there exists a direction l such that the projection of R onto l is the union of n disjoint intervals, where n is the number of connected components of R.
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