DNF tautologies with a limited number of occurrences of every variable

Abstract

It is known that every DNF tautology with all monomials of length k contains a variable with at least \Omega\Gamma 2 k =k) occurrences. It is not known, however, if this bound is tight, i.e. if there are tautologies with at most O(2 k =k) occurrences of every variable. DNF tautologies with 2 k monomials of length k and with at most 2 k =k ff occurrences of every variable, where ff = log 3 4 \Gamma 1 0:26 are presented. This has the following consequence. Let (k; s)-SAT be k-SAT restricted to instances with at most s occurrences of every variable. It is known that for every k, there is an s k such that (k; s k )-SAT is NPcomplete and (k; s k \Gamma 1)-SAT is trivial in the sense that every instance has positive answer. The above result implies that s k 2 k =k ff . This improves the previously known bound s k 11 32 2 k . 1 Introduction Let a (k; s)-formula be a formula in conjunctive normal form whose clauses consist of exactly k literals and such that every variabl..