TITLE: irregular pairs in half graphs - Szemeredi regularity
QUESTION [5 upvotes]: Szemeredi's regularity lemma is a well-known result about partitioning large graphs into pieces such that most pairs of pieces are "regular". The precise statement takes a bit of detail so I'll just refer to the the Wikipedia link.
My question is about the part of the lemma saying "all except $\epsilon k^2$ of the pairs $V_i$, $V_j$ are $\epsilon$-pseudo random". (Note: "pseudo-random" and "regular" seem to be somewhat interchangeable in this context.) Concerning the necessity of these "irregular" pairs, a frequently cited example is the "half-graph".
In particular, given $n>0$, the half-graph of height $n$, $H_n$, is described as follows:

$V(H_n)=\{v_1,\ldots,v_n,w_1,\ldots,w_n\}$,
given, $1\leq i,j\leq n$, $(v_i,w_j)\in E(H_n)$ if and only if $i\leq j$,
no other edges are present in $H_n$ (so $(\{v_1,\ldots,v_n\},\{w_1,\ldots,w_n\})$ is a bipartition for $H_n$).

The graphs $H_n$ were the first observed example of where irregular pairs in Szemeredi's Lemma are necessary. I have two questions about this.
1) What is the precise statement of this claim about irregular pairs in half graphs?
2) Is there a source where this claim is explained in detail?

Remarks:

Concerning question 1, the closest thing I have found is Section 2.1.2 of this survey paper, which says:
"The reader is invited to prove that, for small enough $\epsilon>0$, any $(\epsilon,k)$-equitable, $\epsilon$-regular partition of $H_n$ requires at least $ck$ $\epsilon$-irregular pairs, where $c=c(\epsilon)>0$ is some constant that depends only on $\epsilon$."
But I am still uncertain on exactly how $n$ and $k$ are being quantified in this statement. My best guess is:
"For any small enough $\epsilon>0$, there is $c=c(\epsilon)>0$ such that for all $k$ there is some large enough $n$ such that any $(\epsilon,k)$-equitable, $\epsilon$-regular partition of $H_n$ requires at least $ck$ $\epsilon$-irregular pairs."

Concerning question 2, I have only found sources that mention half-graphs as examples of irregular pairs. So it would be nice to know if this is worked out in detail somewhere.

REPLY [2 votes]: I had been stuck on this same problem for a while as well and asked Dr. Rödl about it. He gave me the following solution:

Suppose for $1/4>\epsilon >0$, we get $M$ as the upper bound on the number of partitions by applying the regularity lemma. We can take $N = kM$ and define the Half Graph $H_N$ that you mention here.

An equivalent version of the regularity lemma allows us to form equipartition on each of the halves (call them $V$ and $W$ and denote partitions by $V_i, W_j$), such that $|V_i| = |W_j| = k$ for all $i,j$

Delete the top and bottom $\epsilon k$ elements from each $W_i$ to form $W_i'$. Do the same to form $V_i'$. Then we delete $4\epsilon kM$ elements at most (at most half the number of total vertices). Then consider $W'$ as the union of $W_i'$. And similarly $V'$. Find a vertex in $V_i'$ called $v'_h$ and select $j$ such that $w'_h$ in $W'_j$ is below it (on re-indexing both $V'$, $W'$).

Then consider $(V_i,W_j)$. The part of $V_i$ below $v'_h$ with the part of $W_j$ above $w'_h$ is a 0 density graph. While the part of $V_i$ above $v_h'$ with the part of $W'_j$ below $w'_h$ is a density 1 graph. Both of these parts are not too small ($>\epsilon k$) in size because of how we constructed $V', W'$. Thus we have that $(V_i, W_j)$ is an irregular pair.

For every $V_i$, we can pick some $v'_h\in V_i'$, and find an irregular pair. Thus we have at least $M$ irregular pairs.