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**Latest Paper Draft (About 1.4MByte)**

**Links to the first version**

**Paper Verson 1.11 , ****Explanatory Notes Version 1.11 , ****Overview of the Shift Method (1.086MByte)**

**Google Docs for extra explanations to the 1st version**

Sections 2 and 3

**l-Extension, ****Counting Marks, ****Double Marks and Lemma 2.5, ****Extension Generator Theorem 1**, **Definition of Extension Generator****, ****EGT2 **, ** ****EGT3 **,** **** ****EGT4 **,** ****Split y of [n], ****Relation to Sunflower .**

Section 4

**DNF and Terms I, ****DNF and Terms II, **** ****Preprocess,**** **** ****Quadruples σ,**

**BlockedEdges I,**** **** ****BlockedEdges II, ****BlockedEdges III,**** **

**LocalShift, ****Lemma by Lemma,**

**Edge Direction Problem, ****Regular σ Only.**

Section 5

**Clique-Tail Pairs, ****Fixed Point of (z, z’), ****First Two Steps, ****Space-Proportional Splits, **** ****Step 3, **

**Links to Hamming Space papers**

**l-extension Ext(U, l) becomes a majority when m is small and U is not too sparse ** (Congressus Numerantium, 173(2005), 33–41)

**Hamming Distance between Two Not Too Sparse m-Set Families is O( m / exp(-ln**^{2} ln m) ), (Full version of the paper posted on 8/26/12. It improves the bound from O( m / ln m ). )

**The sparsity of l-shadow of U is small as l gets smaller than m** (Congressus Numerantium, 168(2004), 21–31. The definition of Ext(U, l) is slightly different in this old paper. It’s due to my try-and-errors.)

**Some basic claims related to the Hamming space ** (Congressus Numerantium, 161(2003), 41–63. )