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First let me note that the lower bound given in the problem should be $c \log n/\log \log n$ because the product of the first $k$ primes is $\exp((1+o(1))k \log k)$ by the prime number theorem. Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107), and page 8 of http://hsb.org.hu/~p_erdos/1981-21.pdf .

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .

Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107; in fact he gives an explicit constant here that he says he cannot improve), and page 8 of http://hsb.org.hu/~p_erdos/1981-21.pdf .

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .

First let me note that the lower bound given in the problem should be $c \log n/\log \log n$ because the product of the first $k$ primes is $\exp((1+o(1))k \log k)$ by the prime number theorem. Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107).

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .

First let me note that the lower bound given in the problem should be $c \log n/\log \log n$ because the product of the first $k$ primes is $\exp((1+o(1))k \log k)$ by the prime number theorem. Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107), and page 8 of http://hsb.org.hu/~p_erdos/1981-21.pdf .

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .

First let me note that the lower bound given in the problem should be $c \log n/\log \log n$ because the product of the first $k$ primes is $\exp((1+o(1))k \log k)$ by the prime number theorem. Erdos has mentioned this lower bound in several places, adding always that he's never been able to improve it. For example see http://renyi.hu/~p_erdos/1951-13.pdf (page 107).

A standard Borel-Cantelli type heuristic suggests that the gaps should be bounded by some constant times $\log n$, analogous to the Cramer conjectures for gaps between primes. I don't know if anyone has written down such a conjecture in this context. But I did find a paper by Kevin McCurley where he considers the least square-free number in an arithmetic progressions, and formulates a Borel-Cantelli type conjecture in this context. In analytic number theory, the situations of arithmetic progressions and short intervals are usually very similar, and one could adapt McCurley's argument to write down conjectures in the short interval case. McCurley's paper is here: http://www.ams.org/journals/tran/1986-293-02/S0002-9947-1986-0816304-1/S0002-9947-1986-0816304-1.pdf .