This randomized phase II trial studies how well cetuximab with or without tivantinib works in treating patients with head and neck cancer that has come back (recurrent), has spread to other places in the body (metastatic), or cannot be removed by surgery. Monoclonal antibodies, such as cetuximab, may interfere with the ability of tumor cells to grow and spread. Tivantinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether cetuximab is more effective with or without tivantinib in treating patients with head and neck cancer.

PRIMARY OBJECTIVES:

I. Response rate (comparing the cetuximab/ARQ 197 [tivantinib] combination with cetuximab single agent activity).

SECONDARY OBJECTIVES:

I. Continuous tumor shrinkage. II. Progression-free survival (PFS). III. Overall survival (OS). IV. Objectives I, II, and III above, as well as response rates, will be assessed and compared between treatment arms in the subgroup of patients with high mesenchymal epithelial transition factor (c-MET) expression, and/or high c-MET copy number.

V. Single agent activity for ARQ 197 (tivantinib) in patients who have failed cetuximab.

OUTLINE: Patients are randomized to 1 of 2 treatment arms.

ARM I: Patients receive cetuximab intravenously (IV) over 60-120 minutes on days 1 and 15 and tivantinib orally (PO) twice daily (BID) on days 1-28. Courses repeat every 4 weeks in the absence of disease progression or unacceptable toxicity.

ARM II: Patients receive cetuximab IV over 60-120 minutes on days 1 and 15. Patients who fail cetuximab as a single agent may receive single agent tivantinib PO BID on days 1-28. Courses repeat every 4 weeks in the absence of disease progression or unacceptable toxicity.

After completion of study treatment, patients are followed up for 5 years.

Inclusion Criteria:

• Histologically/cytologically confirmed diagnosis of squamous cell carcinoma of head and neck origin not amenable to curative intent therapy; both human papillomavirus (HPV) positive (+) and HPV negative (-) are eligible, but status has to be known prior to randomization (although not required for consenting); any type of tissue based HPV assessment is acceptable (e.g. p16 immunohistochemistry [IHC] or HPV in situ hybridization [ISH]); if local HPV testing is not available slides can be sent to the University of Chicago for HPV testing; please note that p16 IHC is generally only considered to be accurate for oropharyngeal tumors
• Presence of measurable lesions (as per Response Evaluation Criteria in Solid Tumors [RECIST] 1.1); generally a >= 10 mm tumor lesion (in the longest diameter by computed tomography [CT] scan) or a lymph node >= 15 mm (short axis) is considered measurable disease when evaluated by CT scan (with a slice thickness no greater than 5 mm)
• Availability of tissue (10 tumor containing formalin-fixed, paraffin-embedded [FFPE] slides/sections)
• Eastern Cooperative Oncology Group (ECOG) performance status of 0, 1
• Patients who have received cetuximab or another inhibitor of epidermal growth factor receptor (EGFR) in the curative intent treatment setting (e.g. with radiation or during induction chemotherapy [prior to definitive, curative intent therapy]) are eligible for the study
• Life expectancy of greater than 8 weeks
• Hemoglobin >= 9.0 g/dL
• Leukocytes >= 3,000/mcL
• Absolute neutrophil count >= 1,500/mcL
• Platelets >= 100,000/mcL
• Total bilirubin =< 1.5 x institutional upper limit of normal
• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 2.5 x institutional upper limit of normal
• Serum creatinine =< 1.5 x institutional upper limit of normal OR creatinine clearance >= 60 mL/min/1.73 m^2 for patients with creatinine levels above institutional normal
• Patients must be able to swallow ARQ 197 (tivantinib) by mouth, unless adequate data about administration by gastrostomy (G)-tube becomes available; tablets may be crushed, but must be taken orally
• Human immunodeficiency virus (HIV)-positive patients with normal immune function (cluster of differentiation [CD]4 count > 200) are eligible if there are no drug interactions with ARQ 197 (tivantinib) or cetuximab
• Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of ARQ 197 (tivantinib) administration
• Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

• Patients who have had chemotherapy or radiotherapy within 2 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 2 weeks earlier
• Nasopharyngeal tumors that show lymphoepithelioma histology
• Patients who have received more than 2 prior cytotoxic treatments in the palliative treatment setting are ineligible
• Patients who have received treatment with an EGFR or MET inhibitor in the palliative treatment setting are ineligible
• Patients with known, active brain metastases should be excluded from this clinical trial; patients with treated brain metastases stable for >= 12 weeks are eligible; use of corticosteroid (for patients with brain metastasis and other indications for corticosteroid use) is acceptable on a low maintenance or tapering dose schedule
• History of allergic reactions attributed to compounds of similar chemical or biologic composition to ARQ 197 (tivantinib) or cetuximab
• Concurrent life-threatening diseases: patients with diseases which with reasonable certainty do not limit life expectancy to 12 months or less are eligible; assessment of such concurrent illnesses should be by the principal investigator
• Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
• Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with ARQ 197 (tivantinib)
• Concurrent use of warfarin (therapeutic use) is allowed, but requires close monitoring of prothrombin time (PT)/international normalized ratio (INR)
• History of congestive heart failure defined as class II to IV per New York Heart Association (NYHA) classification; active coronary artery disease (CAD), clinically significant bradycardia or other uncontrolled, cardiac arrhythmia defined as >= grade 3 according to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 4.0, or uncontrolled hypertension; myocardial infarction occurring within 6 months prior to study entry (myocardial infarction occurring > 6 months prior to study entry is permitted)
• Patients may not be receiving any other investigational agents