Examples of Λ(4) sets E and a graph structure in E x E

We construct examples of Λ(4) sets E ⊂ ℤ. The construction uses certain families of thin intervals ${\displaystyle \left\{I_k\right\}=ℒ\approx E.The\Lambda \left(4\right)\propto ertyf\text{or}Eisobta\in edomthestron\ge rres\underline{t}t\hat{}}$||f||_{4} ≤ c||(∑|f_{I_k}|^2)^{1/2}_||_{4}${\displaystyle where}$f̂${\displaystyle is\supset p\text{or}tedon}$(⋃ I_k) ⋂ ℤ${\displaystyle \text{and}}$f_{I_k}${\displaystyle isdef\in edby}$f̂_{I_k} = χ_{I_k}f̂!$!. The proof of the latter involves a graph structure defined in terms of ℒ × ℒ (which is essentially E × E).