1. C_5 certificate: labels valid=True, 2-iatlon=True, constrained K_3 found=False, triangle=False, independent 3-set=False
2. choices on 5 points without constrained K_3: 2124 of 59049 (paper: 2,124); on 4 points: 69 of 81
3. Lemma 1 graph k=1, b=1: b-iatlon=True, no constrained K_(1,1)=True
3. Lemma 1 graph k=2, b=1: b-iatlon=True, no constrained K_(1,2)=True
3. Lemma 1 graph k=2, b=2: b-iatlon=True, no constrained K_(1,2)=True
3. Lemma 1 graph k=3, b=2: b-iatlon=True, no constrained K_(1,3)=True
3. Lemma 1 graph k=2, b=3: b-iatlon=True, no constrained K_(1,2)=True
4. r(2K_2, K_3) = 5; 2-iatlon choice graphs on 5 vertices containing a constrained 2K_2: 0 of 59049  => p(2K_2,2) >= 6 > r(2K_2,K_3) = 5; random choice graphs on 6 vertices lacking one: 0 of 3000
5. Patak Thm 1(1) for K_3, b=2: 9 (paper: 9); new bound 1+b+b^2 = 7 (paper: 7)
5. Patak Prop. 2 value: 562 (paper: 562); new p(K_33,3) bound 3b^3+b^2+b+1 = 94 (paper: 94)
5. K_{m,n} recursion closed form n b^m + sum_{j<m} b^j: OK for b<=6, m,n<=5
5. K_n recursion closed form sum_{j<n} b^j: OK
5. v+(b-1)e for K_33 = 9b-3: OK
5. Helly = Radon - 1 = sum_{j=1}^{d+2} b^j: OK
5. Patak Thm 2: b*S_{d+1}+1, degree 1+2(d+1) = 2d+3 in b: OK; Thm 4: 2x^3+... with x=C(b+1,2): degree 6: OK
5. NOTE: Patak Thm 6(4) with m=n=3,k=g+1 gives 2x^3+bx^2+bx+b+1+3g(b+1); his Thm 4 states the same minus 1 (a +1 slip in Patak, irrelevant here)
5. surface b=2 g=0: Patak Thm 4 = 80, new = 31
5. surface b=2 g=1: Patak Thm 4 = 89, new = 46
5. surface b=3 g=0: Patak Thm 4 = 561, new = 94
5. surface b=3 g=1: Patak Thm 4 = 573, new = 118
5. (n-1)b+1 for n=3,b=2: 5 (paper: 5)
5. Patak Sec. 5 predicts 3b^3+b^2+b+1+g(b+1); with single-label stars Prop. 1(9) gives g(3b+3); the paper gives g(9b-3): the paper's Remark 5.5 is correct that the linear term differs
6. (k,b)=(2,2), N=5: 200/200 random choice graphs -> constrained K_(1,2) returned and verified
6. (k,b)=(3,2), N=7: 200/200 random choice graphs -> constrained K_(1,3) returned and verified
6. (k,b)=(4,2), N=9: 200/200 random choice graphs -> constrained K_(1,4) returned and verified
6. (k,b)=(5,2), N=11: 40/40 random choice graphs -> constrained K_(1,5) returned and verified
6. (k,b)=(2,3), N=7: 200/200 random choice graphs -> constrained K_(1,2) returned and verified
6. (k,b)=(3,3), N=10: 40/40 random choice graphs -> constrained K_(1,3) returned and verified
6. (k,b)=(2,4), N=9: 200/200 random choice graphs -> constrained K_(1,2) returned and verified
6. (k,b)=(2,5), N=11: 40/40 random choice graphs -> constrained K_(1,2) returned and verified
6. join (Lemma 4.1) for K_3, b=2, N=7: 100/100 random choice graphs -> constrained K_3 built and verified
6. join (Lemma 4.1) for K_3, b=3, N=13: 25/25 random choice graphs -> constrained K_3 built and verified
done in 24.9s; C5 and Lemma-1 checks PASS
