Let be a polynomial ring over a field and let be a standard graded -Algebra where is a graded ideal of . The Multiplicity Conjecture of Herzog, Huneke, and Srinivasan which was recently proved using the Boij-Söderberg theory states that the multiplicity of is bounded above by a function of the maximal shifts in the minimal graded free resolution of over as well as bounded below by a function of the minimal shifts if is Cohen-Macaulay.
We present results to the related problem that the total Betti-numbers of are also bounded above by a function of the shifts in the minimal graded free resolution of as well as bounded below by another function of the shifts if is Cohen-Macaulay.